Finishing the exercises of yesterday first:

This question:

Here I tried convert to normal line form and that made the exercise immensely confusing. I guess it is important in this kind of situation to make sure that I don-t transform initial equation into something else and then completelly forget about initial… the initial form might be in the already most useful one.

Check this question:

 

 

I was totally blocked after 7min… and then suddenly it dawned on me. The main biggest issues was… first of all… that I forgot one can square two sides of the equation… again this is something I don’t have as interiorized as the other rules… I remember yesterday being a bit surprised by it. Second of all… once I realized one can square both sides… then also I don’t have the relationships of (a+b)(a+b) = a^2 +2ab+ b^2 fully interiorized either… otherwise it’d have been easy to see what the solution was. On minute 7 once I realized what was going on, I did (x-y)^2 and of course one can then easily substitute and find that answer is E .

This question

I am quite happy because this was harder than the question 2 questions back, and uses not only the method I learned there but then the expansion as well. I am quite happy because I think I followed the right approach flalessly… but still it took me 2:30 min to solve. 🙁 But it’s a hard level one so I guess it makes sense?

Now check this one:

Again I solved correctly but took me 4minutes. Hmm… reason is mainly that I am not quick enough … sure these are the hardest questions but still… Also the paper and pen one uses matters… It’s important to be fully comfortable when answering the questions.

Check this question:

This one tok me 3m 15s to solve correctly. Hmm there was a quicker way :S . I could have factored things out. Taking the biggest one in each side yields a simplified 2^4 * (-1) which is very easy to solve.

Check this question:

Again I am happy I got right answer but it took me 3m 20s.

Now… I am starting to think that if I wanna get very high score… precissely what I need is to be able to answer these hardest questions in 1min 50 s. Maybe this is really what it takes to kill it… maybe the best people aren’t that much smarter… they are just able to answer these questions like me but in half the time.

Check this question:

This one totally blocked me… hmm… main problem was not realizing how to factor that quadratic polynomial. Actually I could have used the normal method I used to use  and it would have been fine. assume x^2  + mx +c = e and do the usual. When doing that then the equation is factored to (x – y)(x + 5y) = 15 . Realize that that’s the same , just written in different form.

WOA Alarm sounded… thats 1:30 hours gone just like that. Will put 1:00 more

This is my progress so far. Not bad:

So now we go to “Word Problems”.

Word problems

Ok before starting I just lost about 30 min watching some unrelated videos, not necessarily bad, but I feel I hadn’t done enough study to deserve a break XD! So anyway I just put 1 hour on the clock.

Consider this question

Now the issue is… I am getting all these questions right… but the issue is it takes me 2m 30s or so to solve (for example the one above). And the process followed on the video is pretty much what I am doing :S so basically it seems I just need to be faster overall… but if I am too fast I might make more silly mistakes. Let’s see… I hope that the more questions I do the better I’ll do.

Very careful if given a ration not to make a mistake when converting to equation.

Ok for example in this question:

Now this I solved correctly in 5 min because I did make the ratio mistake (that’s why I put the note above). But the main issue that I found is solwing me down, is that I am finding 1 variable and substituting in other equation instead of just solving the system of 2 linear equations by doing it like in the picture above. That would definitely save me time. So next time I will try that approach if it fits and see how it goes.

Ok awesome. So I did 3 more questions of the kind, in this case involving ages, and all I solved in 1m 30s approx by using the method on the picture above. It is important to have that in mind when doing the calcualtions… that one is dealing with a simple system of linear equations thats all.

Now rate questions are coming etc… time it takes if moving at x miles per minute etc. For this, having done some science helps a lot because we know of the importance of the “UNITS” and how they help to find the solutions . This is def. an advantage for someone that has done before physics or chemistry for example.

Alarm sounded 1 hour has passed

Consider this question:

Now I did the whole process correct, it’s just that I made a little mistake when solving an equation :S. Hmm… it’s a bit annoying as the conceptualization of the problem wasn’t easy and I got that right pretty quickly… just need to be careful in my calculations.

Hmm I solved another in 2m48s.

I can see that in the videos he is using a less “logical” approach, and a more “applying formulas” approach. At the end of the day it’s the same… Now, I personally prefer a logical approach not to have to memorize many things… however if I see that the time loss is really hindering me in the future, then it might make sense to try to learn the formulas and apply them more directly… but I don’t know… I hope to be able to just apply basic principles and still be able to find an answer.

Alarm sounded puntting 45m extra.

Consider this question:

Now I solved it correctly in 6m 30s which is failure totally for sure… however, important to understand what happened.

I did the process correctly but then the issue was that I got distance to be (150*7)/12. Now… the result I am looking for is that *(60/75). Now, if you calculate the distance, it is actually not an integer! So I was thinking maybe I got something wrong… The key is to realize that one does not need to calculate the distance yet… better to multiply by (60/75) because then you can factorize the 150 to take out the 75 and the calculation becomes much easier.

Distance expanding or reducing

Consider this question:

Now this took me 5 minutes that I was blocked… due to an initial stupid mistake when framing the equations. On min5  I realized and solved the problem in 1:30 after that, but it took me some time to realize that time taken by Y and time taken by B is the same! Again… sloppy although I do find the solution at the end…. Hope it won’t happen again now that I am more aware of the concept.

Besides that, I am feeling very confident I can solve these types of questions without problem, it’s just that it’s taking me a bit longer than I’d like to. I wonder if that’s what it takes to get a supe rhigh score… to not just be smart and know the fundaments… but just to know the formulas and know how to apply them quickly to get answers and move on. :S That wouldn’t be fun really.

Update: Ok now that I see the answer (which is what I got), I do see there was a smart quick way to tackle this problem even better than doing the formulas. It’s easy, just check how much the distance has reduced between the two people in 1 hour, and there u go… you just need to divide total distance by that “shrinkage rate” and that’s it. Ok, this is good to know …. in case I encounter a similar question this can definitely help.  For expanding is also easy. Find rate and apply by time.

Again check below:

Now the very sad part is that I did everything correctly but again a stupid calculation mistake I made when mentally calculating confused the heck out of me. Basically a mental calculation where 36 /3 = 13 instead of 12.

Now, I want to stop for a second (Alarm just sounded so 45 went by), and analyze what STUPID MISTAKE Is. Because whatever it is, it is comming up much more often than it should… so I might as well address it. From now on I will focus a lot to understand what each “stupid mistake” is when I make it. In this case it is clear… a faulty mental arithmetic. 36/3 should clearly be 12 not 13… Now… I can either try to avoid mental calculations for most cases for example and see if that improves my ratios… or smth like that. Let’s see, for the moment what I will do is to be VERY AWARE every time I decide to make a mental calculation, regarding the risk it entails. Let’s see if this slows me down a lot or not.

Ok, unlike the previous days, it’s now 1am so I am gonna stop here and go to sleep at 1:30 the latest. This way tomorrow I won’t have to wake up too late and I will aim to already gain some work hours in the morning, which I have not been able to do much the previoys days.

I am halfway through this section. Then I will only have 4 sections left… which are coincidentally the very toughest ones for me. Let’s see how that goes. I have not even opened the GMAT official book yet… I really wonder how will I perform with their questions. Let’s see… so far this program seems execellent from my point of view and just what I needed. After finishing the program I will then probably do a mock Quant test to see what happens.

I need probably 3 days more to finish the rest of the sections… it’ll be 7 dec by then or so? Hmm… Let’s see… maybe I can finish next sections in less if I am very diligent. Will try.

Night!

 

 

 

 

 

 

 

Ok so I am gonna do 1 hour to warm up.

Powers and roots

First thing I didn’t remember is that any non 0 number to the power of 0 = 1. Noted.

Odd exponent preserves the sign of the base. If even, always positive result. (as long as the base different of 0).

Suggests to memorize some basic powers:

Hmm… won’t do now unless needed later

Trick for squaring numbers ending in 5. This I didn’t know but anyway I learned just in case. Basically any number ending in 5 its square can be easily calculated by taking nums in front of the 5 , multiplying it by itself+1 and then putting 25 after that. So for ecample 235^2 is (23*24) = 552 and put 25 after hence 55225.

Exponent laws

If same base product with diff exp, u can add. If divide u can substract. If power to a power we can multiply the exponents. This I remember from high school.

Most I know but something I have not used much and didn’t remember is the combining bases law that says x^n * y^n = (xy)^n. So for example 2^3 * 7^3 = 14^3. Not sure when this will be handy but interesting. Actually it makes sense… cause if I deconstruct an integer into its components such as 30 (3*5*2)… then It’s clear to see that 30^2 = 3^2 + 5^2 + 2^2.

Some questions basically ask “what is the units digit of N^big exponent” such as 53^35. Important to know that we need to only do the first step of the multiplication each time, to find the pattern that repeats… then it’s easy to know the units digit of any exponent of 53. CAREFUL … not to forget the N^1 digit in the cycle. I made that mistake in my calculations and gave me the wrong result even doing the method right. For sure I will need a good night sleep before the GMAT or these “stupid

mistakes” are gonna get me in trouble.

Square Roots

square(n) is a number (positive) than when square gives n.

square(0) = 0.

Again suggest to memorize some basics:

These I won’t memorize for the moment but hope to organically do eventually .

square(x^2) = absolutevalue(8)

Also as we know that exponentiating a number between 0 and 1 makes it smaller… the square root of x being between 0 and 1 is greater than x. And the opposite happens when x is greater than 1.

Other Roots

Odd roots. We can find the odd root of a negative number.  The odd root of a negative number will be negative. The odd root of a positive number will be positive

Even roots: We cannot find the even root of a negative number. Even root of a positive numebr will always be positive.

All roots have at most 1 value.

Hmm I can see some trouble coming from not having enough practice with exponents and memorizing. Hope it doesn’t become a problem.

This question:

I solved in 2min correctly. Clearly Iam quite rusty :S. Might need to do some roots manipulation to speed things up.

Equations and Powers

Rule: if b^x = b^y then x = y (except when base = 0 or 1)

This again seems easy.

Fixing the denominator

If answer has a root in denominator one can take it out by multiplying top and bottom by that root. 🙂 Important to keep this in mind!

ALARM misbehaved – stopped and forgot to turn on again. I am not sure how much time has passed… maybe 1:45 min?  Put 2 hours on clock now.

To fix something like below you can use the conjugate. Damn haha this vaguely came to my mind when reading the question. I remember doing this in high school.

Just be careful to know that (a+b)(a-b) = a^2 – b^2 (that – sign is important).

Ok this question below totally got me:

No idea how to solve. Ok after 15 minutes thinking I suddently decided to deconstruct the exponents… and then it made sense… took a long long time, but go it. Hmm… not sure why I didn’t try it before. Mainly I think I am not confident enough doing operations with exponents. Definitely need to practice.

Now in the video this is solved by factoring and exponential equation. Although I am not 100% confident with the system, just knowing that I can expect this kind of operations is already very helpful. Let’s see how I react next time I face a similar problem.

This question

I solved right but took me 5 min, so total failure.

I’m gonna really focus on what went wrong and how to improve.

Ok I think I got it… so the process is to say “I am gonna check different values of x and y that satisfy condition 1. I am then going to check if they satisfy the question target”. And do the same with 2. I think with this method Ill be able to unblock in situations like this question.

Damn damn damn this question also got me :S 🙁

I am not sure at this pace 🙁 how well Im gonna do… I guess as long as I don’t repeat same mistakes 🙁

Here first I failed to realize that x^21 is (x^7)^3. I had done (x^7)(x^7)(x^7)  but just didn’t really see it being the above. Then even after knowing that, there are more hard stuff… now that we know (x^7)^3 = 2^3 the smart thing is to realize that x^7 will be 2, and only 2 because exponent is odd. If it was even then -2 will also work.

On the expression below, we see that x^7 has to equal 2, but the thing is… it could also be -2 and give the same 4 result once squared hence it is not sufficient. DAMN DAMN DAMN I should be sharper. I get all this and know it, but just don’t do it when I am answering the question.

Check this question:

Again didn’t know how to solve :(. Thing is, I did the table thing that make me understand that statement 1 is sufficient, but statement 2 I was not clear. Moreover statement 1 I was not 100% sure I was fully right.

The issue is that this question really is complicated. First of all… x^n = y^n I never “re-wrote this”. If one re-writes it to (x/y)^n = 1 then things become a bit more manageable.

In this situation one needs to be smart enough to understand what are ALL cases in which that equation is  met. This is not trivial. Well it turns out there are 3 cases.  Case 1 is when x/y = 1 and n is ANY value. Case 2 is when x/y is -1 and n is ANY value, case 3 is when x and y are ANY value and n = 0. With this in mind we can go ahead evaluate.

First case tells us x/y = 2 hence already we know it can only be the other case where x and y can be any value and n = 0 hence sufficient. Statement 2 on its own however… tells us that y and x are not the same hence we know that to yield 1 will be impossible so case 1 is not. Then case  yield -1 will be possible with something like 2 , -2 for intance… now, in that case N can be anything, so n surely can be any number less than -2 and still will hold hence we don’t know the value of N. that already means statement is insufficient. Case 3 if we analyze we see that n would be 0, and y and x could be any number… but still is NS because we know N can be many other numbers besides 0.

Learned that problems can indeed get super complex. Cases must be split, I must try to find ways to rearange given equations to make my lfie easier.

Ok I got 3 or 4 right in a row. Even some hard ones like this:

Good… maybe little by little the concepts really are starting to sink in, which is good.

Done with the powers and roots module! Let’s get into Algebra!

Starting all very basic. I have decent algebra basics :)! Again high school does help.

Alarm sounded. Gonna put 2 more hours. I am watching videos quite quickly as quite basic.

Ok so I just got absolutely owned by this:

Certainly I am not thinking straight.

Did it again and took me 1m 52 s to find an answer just by doing a simple substitution by 1 instead of trying to simplify the heck out of the equation.

Now in the video he actually simplifies… in order to be able to do it efficiently there is one thing I had not memorized that I need to memorize which is that (a+b)(a+b) = a^2 + 2ab + b^2 .I just need to be sharp enough to look at an equation and realize if that meets or not.

Question below is hard:

 

Thankfully I took right approach and decided to substitute values for x and y. Got right answer although in 2m 10s, because of making some silly mistakes when calculating

Next question also was interesting:

 

This one again I got right in 2min approx … at the beginning I didn’t know what to do… but then I realized the expression was equal to (352+1)^2  – 352^2 and by expanding that got to the answer.

Actually there is a MUCH quicker way … that is realizing the numbers are a substraction of squares a^2 – b^2 hence we can know that they come from (a+b)(a-b). Which takes us to answer much quicker.

Equations

This is also simple. Went also through quadratic equations. Also sistems of linear equations. Hmm… interesting the “Elimination method” I just needed to refresh it a bit.

Systems of equations can have one solution, infinite or none (if x + y = 2 and x + y = 23 for example). If both equal, then infinite number of solutions.

Systems of 3 equation swith 3 unknowns. Follow similar process. Just convert into 2 by substitution and then solve.

Interesting… I don’t know why this is counterintuitive… but in an equation I can raise both sides to the power of 2 for instance and still meets the conditions. Well I guess it makes sense actually… because if a = b then a^2 must equal b^2 .

Extraneous roots

root(k)^2 = k for k bigger or equal to 0. Careful sometimes exponentiating etc to eliminate roots might give results that are not valid as when substituting u find u are dealing with a negative root.

Equation with absolute values

Here the important thing is to realize that if abs(x) = a then x can be x=a or x=-a . So we can do that whenever we encounter an equation with absolutesl.

Careful: When doing equations with absolute valeu there can again be extraneous roots. So need to check the answer on the original equation and see if it makes sense or not.

Check this question:

It’s a shame because I did the whole process well but made a little mistake in an addition hence I got confused :(. Need to be very careful when doing calculations. It gives me peace to know that the method I was using was right though. Honestly if I hadn’t studied this absolute value section I wouldn’t have been able to answer these kinds of questions.

Remember, all one needs to remember is that the equation can equal either a or – a and then make sure to double check your answers in original equation to see if any extraneous root is there.

STUPID MISTAKES – > Are happening much more often that I would want to. Hmm… I am thinking that still is early on my study so its normal… but I am also thinking it might have to do with the process I am using. I am doing a lot of calculations on my head and I think that’s causing it. So I will try write more on paper and see what happens.

Inequalities

Finally. 😀 I have trouble with these so looking forward to refresh this.

Rules:

• I can substract or add both sides … and it remains intact
• I can multiply or divide both sides by a positive number and it remains intact.
• I can multiply or divide both sides by a negative number but that will REVERSE the inequality

Nice …. so it’s actually like an equation pretty much.

If inequality has 3 parts then we can still do the operations above highlighted.

Try this question

I solved properly in 38s. Nice.

Check this one:

I solved in 3min 14s unfortunately, but I am happy I actually solved it. It was pretty damn hard.

I took correct path by understanding that there are 2 cases, y is positive or y is negative. Still I could have done it even better by just rearranging the inequalities in those 2 cases. What I did was a bit more convoluted.

Combining Inequalities

Rule 1 : Rewrite inequalities facing the same direction before trying to combine them .

Adding Inequalities

Remember sign has to face same direction.

Rule 2: Do not substract , divide or multiply inequalities. It doesn’t work.

Rule 3: You can add inequalities , it works.

Check this question:

Im thrilled to announce that I got it right, although it took me 3min to get it right.

Process wasn’t flawless… but eventually it worked.

In the video solution he uses a much quicker method by adding the inequalities. I understand the concept… let’s see if in future questions I can use it though…

ALARM sounded so 2 hours passed. Ill take 10 min break and will resume after. Hmm… I shouldn’t go sleep after 2am though.

Inequalities and absolute values

Here the main rules

Hmm I have not memorized them but I understand the concept. So if a number is more than an absolute value x it means than X is closer to the center than that number. If a number is less than an absolute value X it means that abs(x) are all values further away from center starting on a.

if I have abs(x) < 5 then I need to complete adding engative number -5 < abs(x) < 5 and then remove abs so -5 < x < 5. Ok I think I got this memorized 😀

If abs(x) > 5 then I need to again do the same -5 > x > 5 however it can’t be both so it will be wither one or the other hence -5 > x OR x > 5.

Quadratic Inequalities (really? XD)

Ok basically one finds the solutions, which are the “0 points” where the funciton cuts the axis. If we found 2 solutions then we know we can divide the number line in 3 segments  (left of first 0, between 0 , and right from right 0). So u just need to test a number in each region to know what sign is that range having and that’s all, you can then find the solution to the inequality.

Check this question:

Unfortunately again I made a very very stupid mistake. I am happy because I did the process right… but sad I made such a stupid mistake. Basically I found of course the breaking points 1 and3… but the thing is… when I substituted 2 into the equation to see the sign… I was convinced it was giving me positive… not sure why… I thought it was 2x instead of 4x … really weird . Anyway It’s a bit late maybe that is affecting.

The something technique

Ok so this is an interesting technique to solve equations. Having said that, I honestly prefer to substitute values, I find that’s pretty quick generally. In any case this technique is defintiely good to know.

Consider this question:

Again pretty sad because I did the process well but got it wrong due to again a STUPID little error in my factoring. Seriously sad.

The coordinate plane

This is it:

 

Hmm this should be easy.

Just need to remember coordinates is (x,y)

Distances in the coordinate plane

To find distance just make a right triangle and with pythagoras theorem know the hypothenuse.

Graphing lines in coordinate plane

Easy. Just find where y is 0 and x is 0 connect.

Slope of a line

Go from one point to another, slope is the ratio of upwards movement by horizontal movement so increment of y / increment of x (Can be negative or positive slope). (a,b) (c,d) the slope is (b-d)(a-c).

Vertical line has undefined slope.

As magnitude of slope increases, line gets steeper.

x and y intercepts

Easy no need to explain.

Slope y intercept form

y = mx + b – This tells us already a lot. m is slope, b is the y intercept! Nice.

Graphs of quadratic equations

Just know poiints that are on parabola satisfy equation. Also points are symetric on an axis of symettry, so same distance from it.

Ok, so all this I knew.

DONE with this module! 😀 ! Now only some exercises remaining which I will do, and I can close this module. This was a 50+ video module. Did pretty quickly as a lot of the basic things I knew, and also I am using 2x speed on the videos.

Ok So I am gonna do 19 practice questions let’s see what I get.

So far got 11 right and 2 wrong… but I am a bit tired so better stop. The hardest ones I can do tomorrow.

Been a good day. Did quite a lot.

 

 

 

 

 

 

Here I am after a very long day involving meetings, and sending reports from home afterwards. Of course I was shattered by the end of it so instead of making the same mistake as yesterday I decided to take a proper 1:30 hour nap (the full sleep cycle) and here I am feeling pretty refreshed and excited for these nest 2 to 3 hours of GMAT study.

Arithmetic

How beautiful this word “Arithmetic” I feel like some old Greek sage about to be enlightened. Let’s do this. There +30 videos in this series. Looking forward.

First explain the associative, commutative properties… no need to explain, is very basic. It is still interesting to see how these stuff one has interiorized from high school. For example the Distributive property means a(b+c) = ab + ac. This is something already we do automatically thankfully 🙂 So it is NOT true one forgets EVERYTHING in education.

Ok this again caught me TOTALLY off guard:

First of all because I guess it’s not “clear” its a Data sufficiency question, maybe if the answer choices will be there I’d have realized… but anyway.

I TOTALLY became overconfident and disregarded this question… obviouslu statement 1 is insufficient cause heart could be division or substraction. Then statement two could be division or multiplication… hence using both statements one finds heart is division? WRONG

I need to be EXTREMELY CAREFUL HERE AND MAKE A VOW to approach DS questions in a particular way, or Ill be in trouble. Let’s analyze if statement 2 is sufficient on its own. Well heart could be division or multiplication right? so 1 heart one… is 1 ! no matter if we use division or multiplciation, hence the statement 2 is ENOUGH and SUFFICIENT to answer the question.

Rule 1: OVERCONFIDENCE WILL KILL ME . Treat EVERY question with maximum care, maxium concentration, or don’t do the GMAT. I vow to never underestimate a question again. 

Then video explains how to deal with different sign etc… again this I know from high school.

Absolute value = is the number’s distance from 0 on the number line. (Hence abs(-2) is 2)

Decimals

Damn, dividing and multiplying decimal numbers is boring :S Anyway, let’s not enter into a negative cycle. Good thing is I refreshed how to do it.

Rounding

If 1 to 5 included then round up.

Multiply and divide powers of ten

Again this is easy, no need to explain.

Consider this exercise:

Now the crazy thing is that again, although this time I did everything right, at the very end I ALMOST really messed up. By saying both convined are sufficient when clearly not.

NEED TO NOT ONLY TAKE QUESTION SERIOUSLY, BUT TAKE METHOD VERY SERIOUSLY, SAY IT OUT LOUD. I VOW TO NEVER MENTALLY DECIDE AN ANSWER AGAIN. BEFORE ANSWERING AND MOVING ON I WILL EXPLAIN WHY THE ANSWER IS WHAT I CHOSE.

Fractions

Basic stuff no need to explain.

Video suggests to memorize this table

You know I don’t like memorizing so I won’t. (Also its quite quick to calculate those by doing a brute force division if required). Later if I find myself in a situation when this would have saved my life… I’ll come back here and memorize.

Decimals to fractions

Lol, I thought this was something I hadn’t done but its obviously not. 0.35… I just multiply by 100 so 35/100 hence 7/20 etc.

Undefined n/0

n/0 is undefinded. If for example 6/0 gave us a result R, then 6 should equal 0*R which is not possible obviously. 0/0 also undefined

Properties of fractions

Hmm… not hard enough to be worth writing here. Basically if you had good high school math you will remember this already.

Just one observation… for example in this question:

Here I just reduced the fraction to 45/11 which clearly 11 is at least 4 times in 45… so then I thought… why didn’t you do it to start with? Sure… If a bit quicker I could have realized that 360/88 is just 360 divided into 88 which just means “how many times is 88 in 360” whose answer is clearly > 4. (although one could get confused with the 3.91 answer… so reducing the fraction to 45/11 makes it more clear that for sure 11 is at least 4 times in 45.). Better to play it safe I guess.

Property: If you increase numerator and denominator by same amount, fraction approaches 1… this makes sense because imagine u add 10000000 in both top and bottom, of course it will approach 1.

Check this question:

Now interestingly it took me 4 min to find some answer which turned out to be wrong and was not able to actually find a quick smart way to find a proper answer… well after looking at the solution it turns out this is complete Bullshit. There is no smart way to do this? Basically the guy just says… ok, remember the table you memorized? U can clearly see that 0.375 is 3 times 1/8 … WTF. Seriously? Ok this is annoying. I mean, GMAT is supposed to test logic etc… not this kind of pointless memorization.

Ok so having found this… I think I will have to kind of memorize the table… let’s try.

First of all, I am trying to memorize all fractions with num 1 and den n going from 2 to 9. Let’s start. 1/2 is obviously 0.5 (I guess one does “memorize” sometimes ;P but I would call it “interiorizing” instead). 1/3 actually it’s 0.333333 and this also I know… should have seen that 0.6666 is gonna approx be 2/3. then 1/4 is 0.25 easy. then 1/5 is 0.2 easy… 1/6 I din’t know but it seems it is 0.166666 indeed… so I will keep that in mind. Easy to remember cause sixes match. 1/7 it seems its 0.14 approx (although more decimals are there if one does the brute force division). 1/8 is 0.125 and 1/9 seems 0.111 approx.

Main takeaway: I could have actually, if I had a colder head, calculated each of the above quickly by doing brute force calculation. I guess what stopped me was the thought that “there must be a smarter solution”. It was unthinkable for me that one had to just know, what each of those decimal numbers translated into… honestly. And just brute forcing a multiplication and division with decimals was out of the question.

So I take the learning and move on.

Percents

Again suggests to memorize table

Is same as before actually… so hmm… shouldn’t be a problem!

Interesting technique:

I like this technique. Saying… ok what number I need to multiply denominator to get 100… answer is a num slighly bigger than 2. So I can also multiply top by a number slightly bigger than 2… hence it is clear that answer is slightly bigger than 22%. Love it.

This question took me 3m 14s to find the answer… because I complicated myself:

I need to be very careful not to mess up easy questions like this. In this case I quickly found out % was 36/150 but then complicated myself simplifying the expression etc… I honestly should have just brute force divided it to find result was 24.

Alarm sounded 2 hours went by. Will put 1 more.

Percents increase and decrease

Here just need to be careful to divide the difference by the original number not the end number. That’s very important.

Here this question:

Now I solved it well in 2m 40 s which is failure, using equations etc. Actually if I had just assumed price initial is 100, then would have been much easier. Solved in 55s using the method suggested by the video.

Also I need to be extremely careful… if pop is 60 and increases by 300%… new population is? CAREFUL . Again I underestimated this and just said 180… WRONG. I am forgetting to add the original to the increment so its 240. Seriously I need to stop rushing or I am gonna have problems. It is true that I am watching videos ar 2x speed and I know this topic I already kind of now so I am less careful… but it is not an excuse.

Simple interest rate

This I didn’t know the formula. Interest = principal* rate * years. But after seeing it it becomes obvious that yes, this is what it is.

Compound interest rate

Now if need to find exact then no way one can do it manually. So definitely MUST know formula for this one.

Above is the formula. THANKS GOD it seems that one doesn’t need to evaluate. That’d be impossible really without a calculator as u can get a number to the power of a very big number. When solving the problem I was wondering what I was missing.

MOREOVER, careful… as the question might specify the compounding hapens quarterly for instance… so c = 4. I missed reading that properly… careful.

IMPORTANT TRICK

If the number of compoundings is small… it might just be easier to calculate the % directly. So in question below:

I tried the formula and got confused… hmm but honestly to do the method above one needs to know that 12% compunded monthly means 1% monthly. This I wasn’t aware.

Ratios

Seems simple enough to not need to go deep into examples. Some tricks are useful however. For example. Bruce has 15 cookies, which he will give to K and P in a 2 to 1 ratio. How many cookies does K and P receive? So here I did the calculation etc… butwould have been easier to realize that… for every 3 cookies K receives 2 and P 1. So for every 3 group of cookies, K receives 2 and P 1. Hence we divide 15 by 3, so easily we see K receives 10 and P 5.

Alarm sounded, will try push 30 min more as almost done with the Arithmetic series.

I am getting a bit tired. Got a ratio question all good but was making mistakes in the basic calculations. Need to be very careful.

Another question hapened the same… I am making mistakes when multiplying basic numbers. I need to practice my basic multiplication

Okafter 1:30 hours more I finished the arythmetic. Not happy fully because its almost 3am. I shouldn’t do this 🙁

Anyway, next time Ill take more care.

 

 

 

Put alarm 2:30 hours.

Data sufficiency overview

Ok so I am gonna tackle all the series of data sufficiency videos at gmatprepnow.com. By the way I like to watch videos at high speed. So I watch usually at 2x speed and find they are still very understandable.

• Important to evaulate each statement independently.
• One very important technique to avoid making mistakes is to write the equation that represents what you are looking for. For example in the question below.

If the equation isnot writen, it is extremely easy to make “silly mistakes”. So I will try to adopt this approach!

Each DS question is either a yes/no question or a value question. For a Y/N question ur objective is to determine wheter the answer to target question is always yes or always no. For a value your goal is to determine wheter there is one unique value that can be deduced using the info provided in the statements.

Now consider this:

Actually I am not in the mood right now, I am distracted because I spent all morning in a bureaucratic process related with the VISA of the country I am in right now, and I know that tomorrow I have a full day of work for something very important I need to prepare.

This is leading me to… 1st, not answer the questions before the video gives the answers…! And this is very bad and very dangerous.

In the question above I actually thought statement 2 was NOT sufficient. Missed something very basic. Definitely this state of mind is not appropriate for studying. I can’t continue like this need to take a little break or a nap or something.

Let me try finish this video. Exercise below:

This one I solved in 2m 6s because I have not yet done inequalities etc. Still managed to find the answer doing the right method… just have to remember that multiplying both sides or dividing doesn’t seem to change the “direction” of the inequality. I remember in high school some operations ( I guess when sign changes or smth) do change the direction. I’ll have to learn those at some point.

Again another question to rephrase wud be “Can the positive integer k be expressed as the product of two integers, each of which is greater than 1?” I thought ok well one thing we know is… is k>4? but I failed to get the deeper point… that if a number is product of 2 numbers greater than 1, then the number for sure WON’T be prime! Again, definitely not sharp now, definitely not.

Same with the next is abs(x) < 1? Well if x was an integer this would mean he is asking if x is 0. But X MIGHT NOT be an integer hence it is asking is -1<x<1 , again, I missed that part, I assumed X was an integer. This is WRONG. Also could have written as … is x^2 < 1? It is EXTREMELY important that I wake up and really start caring about those rephrasings because I can see it is exactly the kind of thing that will come up.

Another example… If x and y are integers and x < 0 , is x^y > 0. Well again I MISERABLY failed to see that even values of y would satisfy the condition.

STOP STOP STOP. I can’t continue like this. I will take a nap, relax and come back in a while. This is counter-productive.

Ok after an hour and a half of napping and rest I am back:

Data sufficiency overview continuing

Ok so main most important point learning so far is that IT IS important to re-frame the question and write a formula if possible to udnerstand what is it that one needs to find. I like this a lot and will definitely do it:

A great example is for example this exercise:

As you can see the question is not immediately obvious. I tried but couldn’t really remember the rules for triangles so couldn’t really solve. I kind of remembered the pythagoras theorem but not fully sure! Actually when you rearrange the equation you realize that they are basically asking is x^2 + y^2 = w^2 which basically is the pythagoras theorem! And that will be the case if angle c is 90. So basically the question is asking… is angle C 90?

There are some more examples but I won’t copy each as anyway most topics have not touched. Now it’s mostly the strategy one needs to remember. I see definite massive weaknesses on geometry etc as I forgot everything. Need to revise that very properly.

Here goes another super interesting example. This one I put because it has to do with integer properties which I have studied:

I need to admit that although I was super close, I didn’t get it right in 2m. I understood the question well, the main problem was that I wrongly assumed that if the integer was higher than 1, then hmmm must have a 2 at least hidden inside. This is stupid. Of course many numbers higher than 1 DON’T have a 2 inside, and the most important thing… PRIME numbers actually only have themselves inside! Which is exactly what this question is asking… is K not prime? Now, I do feel a bit bad I didn’t get this one after all the study of yesterday… but it has help me think for sure and I think it reinforces some of the concepts of yesterday so I am not feeling that bad.

Another example:

This one I did right but took me 3min to evaluate which is way too long. I was honestly expecting some clever method to be able to apply instead of brute force but couldn’t find it so just did brute force. The video also used brute force but with a much much quicker calculation regarding what 4^4 is for example etc.

Another one:

Here I solved well, but again took me 1m 56s to evaluate one statement. I need to be quicker. Little by little !

One more:

Luckily I realized the trick here. I almost almost got it wrong. The key is to understand that when X is positive, this doesn’t mean positive INTEGER. Hence 0.5 can be a value of X hence statement is Insuficient.

Now a MASSIVELY important detail. Check this question:

Now, hmm according to video… in this case if we need to prove YES happens ALL the time, it is not enough by just checkign 5 nums… if truly it is true that YES happens always there must be a mathematical proof. And in this case, although common sense already shows that if I do 0.5 or 1 or 100  it will always be yes, there is a way to actually see it clearly. If one substracts 6x from each side of the equation then we end up with is 0<x (is x positive)? And this is clear that, if x is positive, for SURE always is YES. Hence sufficient.

I need to definitely practive inequalities a lot. They come up so often it seems.

Hence table is good to demonstrate insufficiency, but only good to HINT sufficiency!

Here another interesting example:

Again tried to rearrange. After 2 min concluded sufficient, however some seconds after chosing it, I realized about 1!!! tricky, so its NOT sufficient, which is the right answer. I need to be EXTRA careful. I know the concepts and can see the tricks but need to really be careful. In this case the trick was realizing that when I divide 1 by 5 , the remainder is 1! Because the result is 0 with remainder 1. Luckily I had studied this yesterday.

Another important thing… although true some of these I am takin 2 min, sometimes the answers take 20s or so if extra easy and those I don’t really put here. So even if on exam I go a bit over time, I hope that other questions will help balance. An example of these questions is this one:

This one I solved in 15s or so. Basically my thought process was… k +1  means a number to the right of a prime number. So immediately I thought… most prime numbers are not next to each other, but some are, like 2,3. So testing those 2 already shows that statement 1 is insufficient.

Extremely important when testing numbers not to forget negatives, 0, fractions etc.

Geometry DS

1. The picture doesn’t mean anything, don’t estimate from the picture
2. To find one length we must be given at least one other length. (This is very important to remember).

For example the image above does not give any length hence we can’t know AD no matter what.

Now this is extremely tough:

I can see how this is complicating quickly . Basically the video says that as both statements together FORCE AB to have only 1 length , and we are given one length, that means we can know the length of AB. Now, no need to calculate it, but we know it can be calculated.

Although I trust this guy etc… I wonder if that length can actually be calculated. Maybe once I do geometry I will find out, but hmm… I will be very confused if I find that even using geometry rules one can’t find the exact value of AB, cause in that case both statements togethers shouldn’t be sufficient right? The question answers what is the length of AB. Hence we should be able to provide an answer. My intuition tells me that yes, there is a way to calculate it, we will see in the future.

Statements can’t contradict each other

This is important to know. Also I see I need to study again quadratic equations, how to factor them etc. It is very important.

Tips

• Do not assume numbers are integers
• Do not assume numbers are not integers. (example below shows how tricky this gets)

Here u can see an extremely tricky question. It would be easy to think that both statements together are not sufficient… but if one realizes that there can’t be 1.9 boys or something like that, one sees that with both statements together, the only number of students that there can be in the class is 10, hence the statement is sufficient.

• Real word questions might imply positive integers. Careful (Age etc…)
• Do not over calculate.
• Not all equations with 1 variable x and no powers greater than 1 have just 1 solution. The equation can actually be one in which x can be any number, hence not sufficient.
• Not all quadratic equations have 2 solutions
• Not all equations with 2 variables can’t be solved when other conditions given (when a condition is given , for example x is positive integer, so can’t be 0)
• Before starting to examine one of the statements, look at both 1 and 2 and decide which statement you want to start evaulating first (the easiest one)

Factoring quadratic equations

Ok so I don’t wanna ignore some of the weaknesses so I have started reviewing some stuff. Now one intereting and very useful thing I can maybe use is “factoring quadratic equations” by inspection. Now this only works if the roots of the equation (the solutions) are real. I don’t think GMAT tests imaginary numbers XD so I am assuming I will be able to use this in the exam if required without problem. it helps because it really does help in allowing one to decompose stuff into simpler terms.

The best explanation I have found is this video. Awesome. Love this new skill acquired.

Solving quadratic equations

Also just in case I have refreshed the formula of the Quadratic equation for an emergency if required:

Damn this is bringing so many memories back XD! used to love math in high school.

Ok so dedicated some time to factorize quadratic equations and pretty happy now.

ALARM SOUNDED, And added a bitmore thats almost 3 hours of solid good study. That nap really helped.

Conclusions of Day 3

Today I learned quite a lot about Data Suficiency, this should definitely help for the future study.

Also refreshed quadratic equations and factoring them.

Then most importantly, I re-affirmed that it is pointless to study in the wrong state of mind. It is just a waste. The nap method worked great. After taking the nap I was able to make about 3 hours of very solid study.

Todos

1. Improve my brute force multiplying and root skills
2. Geometry will be tough, will have to prepare well

 

 

 

Day 2 of GMAT Preparation

Divisibility Rules

I had encounteered these indirectly on my 1st day of review yesterday and DEFINITELY are required in order to solve some questions quickly.

Rules are decently easy to assimilate:

Divisible by 2 if it is even num, divisible by 3 if sum of digits is div by 3, by 4 if num composed of last 2 digits is div by 4, by 5 if ends with 5 or 0, by 6 if div by 2 AND 3, by 9 if sum of digits div by 9, by 10 if ends with 0.

Now, question below took me 1m 56s to solve, which is totally insufficient especially given its easyness. Why it took me so long? Because I didn’t understand the question. I thought the 1) and 2) where the “characteristics” I had to test with or smth, I thought I had to confirm if given that K is positive integer less than 6, then those two conditions match, which is not what is being asked. Those 1 and 2 are actually clues. So basically you know that k is a positive integer less than 6 (ALSO I had to check after but I wondered… is 0 a positive integer? NO. Positive integers are greater than 0), and you know that 3 is a divisor of N, and N is a multiple of 4 (which basically means 4 is a divisor of N).

At about 1min I realized and that’s when I solved it in 56 sec.

Moral of the story… CAREFUL when reading questions. Not pre-assuming things too quickly. Just relax and focus and read what is being asked.

 

 

 

 

Primer numbers

Video suggests memorizing all primer numbers until 59. I am not sure how important is this. I don’t like memorizing. Some important points I learned are:

1. 1 is not a prime number
2. Prime numbers have exactly and only 2 positive divisors
3. 2 is the only even prime number

Instead of memorizing the primes until 59 which again memorizing is very rarely a good idea, I have developed a very interesting exercise. As I know all primes are odd except 2, and I know by heart that 2,3,5,7,11,13 are the first ones… then I just need to quickly keep checking each subsequent odd number, and discard the ones that have obvious extra divisors. I did that very quickly and gave me the list below. Very interestingly I wrote 4 numbers wrong. Numbers I quickly thought were primes but are not, this is SUPER interesting because it shows where my weakness is.

2,3,5,7,11,13,17,19,23,27,29,31,37,41,43,47,49,51,53,57,59

Now, let’s analyze what went wrong:

• 27 – Here I just wasn’t careful. Obviously if I sum 7+2 = 9 hence number is divisible by 3 and 9.
• 49 – Tricky one… as it doesn’t conform with 2,3,5 which are the common ones I look for. 49 is divisible by 7 as it is 7^2. Tricky one indeed and very improtant I don’t make this mistake again.
• 51 – Here I just wasn’t careful, adding 5+1 = 6 so obv divisible by 3. Tricky anyway cause it is 17*3, so that prime 17 messes up a bit the perception that’s why i missed and why it is so important to know the divisibility rules
• 57 – Again, adding digits gives us 12 whcih is divisible by 3. Same case… 19*3, so that 19 prime embedded in the number is what confused me.

Now without looking I will write the list again:

2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,51,53,59

Again that careless 51, but much better. Now, what I WILL memorize to make sure I don’t make a careless mistake is that until 59 there are 17 (which is prime!!!) prime numbers. This will ensure I don’t make mistakes again.

Hence if someone asks me, what are all primes till 59, I know it’s 17, and don’t need to use memorizing which is SO unstable and prone to errors. I can just easily deduce it: 2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59

Doing this exercise prompted the following question: Is there any odd number till 59, whose sum of digits is NOT divisible by 3, and is NOT a prime?

3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,41,43,45,47,49,51,53,55,57,59

And the answer is… only: 25,35,49,55 and out of those, it is obvious which ones are divisible by 5, and we know 49 is the only super tricky one. HENCE, one can SUPER quickly and with confidence find all primes till 59 by quickly checking each odd number if it is divisible by 3 or not, (and just making sure the obvious divisible by 5 don’t leak, and that 49 also doesn’t leak). If sum of digits not divisible by 3, then its a prime.

2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59

(Also indirectly by doing these exercises I am starting to indirectly memorize the first 8 primes with confidence and without having to think, 2,3,5,7,11,13,17,19.

As a last exercise let me try do it till 100 see what happens:

2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,91,97

So it turns out I didn’t do that bad… I doubted about 67, and then I made a mistake with 91. Why? because 91 is not divisible by 3, but again is one of those tricky 7*13 numbers coming from primes. Definitely the table of 7 causes confusion. Almost made a mistake with 77 too but that was easy to spot.

• There are 17 primes till 59
• There are 25 primes till 100.

With this exercise in mind, I am pretty confident about this topic, although it is 80% confidence. I shouldn’t make ANY mistake. Hence it is not yet perfect. But I will keep training it.

Factorization

I love factorization. It is like “de-constructing” a number into its unique components. And this although I knew it more from inutition than from formality is precisely the “prime factorization theorem” that states that any integer greater than 2 is either a prime or can be expressed as a product of prime numbers.

Deconstructing 85 = 5* 17

Deconstructing 36 = 2*2*3*3

etc. This is also the way we find the positive divisors of a number as we analyzed in previous day. So 85 has 4 positive divisors, 36 has 9 positive divisors. I know already without calculating thanks to the formula we learned… Very very careful here, because the formula is a product not sum. So for 36 it is 3*3 not 3+3, I actually made that mistake on the first time. but if you want to test…:

36 – [1, 2 , 3, 4, 6  9, 12,18, 36]

If Y is a divisor of X, then it means Y should be “hidden” somewhere in the prime factorization of X

Next is the exercise below:

 

Pretty happy because I solved in 1m 6 sec, and not only solved but said that there are 3 different prime factors, and 24 total positive integer divisors.

Now, obviously I am getting good at this. HOWEVER, I did doubt for a second regarding the wording. It says “How many different prime factors”. The factorization is (2^2) * (5^3) * 17. So what do they mean by “different prime factors”. Well, 2*2*5*5*5*17 hmm… there are 6 total prime factors, out of which 3 uniquely different. Wording is still not 100% clear to me TBH and that is a but annoying, but anyway, luckily I understood correctly. “different prime factors” means that… how many of the prime factors of the number are unique, or different from each other.

Next exercise is below:

I solved it in 2min 10s, not ideal at all… Again, in this case I can see that familiarity with the kinds of questions asked is going to be extremely important. That initial facing the totally unkown makes one waste some valuable seconds.

Now, I did understand the question and proceeded to factorize each number in that 2 to 20 even inclusive range. I made a little mistake and counted 19 “2’s”, when actually there are 18. In any case I chose the right answer as I was confident with the approach, but I need to for sure be careful in each calculation I make because these silly mistakes are exactly what ends up killing me. If 19 had been in the answer choices I’d have had a hard hard time.

Imagine the num was 8, then 2*2*2, then the higherinteger n so that  2^n is a divisor of P would be 3. Keep that in mind too.. because 2^3 is 8 and 8 is a divisor of 8.

Finally, best method to avoid mistakes when factoring? Well… it’s a bit of an advanced topic really, but Ideally if I had infinite time I could devise a strategy. Overall what I am seeing is… if number is big, you wanna make it small asap. Ideally if 10 (2*5) is a divisor, take it down asap using that, also if multiple of 5, do a normal division by 5 to take it down, as otherwise it can be confusing, then take it down by halving it and getting rid of all 2’s and you will probably be left with 3’s and some bigger prime number maybe. Anyway, by doing more and more of these exercises in the future I will anyway be getting better at precision and quickness.

Squares, perfect squares

Squares of integers are also known as perfect squares.

Rule 1: The prime factorization of a perfect square will always have an even number of each prime

Now, I feel this is the kind of thing that is a bit of a “curiosity” but that if one doesn’t know, then if it gets asked, you are screwed. For example If I get asked a question like this (this is another video below ( When I study a topic I like to search different sources for different explanations helps a lot).

Then if you don’t know that rule, you will be in deep trouble. However if you know the rule, then you know very easily than I am gonna need at least one more 7 and one more 5… to make this work, and then add any other pair of primes or different pair of primes to find other numbers. Hence I just need to look at the given possible multiple choice answers (in this case u can’t see in the image but it was 5,70,105,140,350), and then see which one of those has a 7 and a 5, and then pairs of another thing. 140 is the one (7*5*2*2).

RULE 2: A perfect square will always have an odd number of positive divisors

Ok, this is getting a bit ridiculous! XD. Now, having studied pretty well the previous lessons, this makes total sense for sure. if I am always gonna have pairs of factors , then it means that the number of positive divisors will always be (odd) * (odd)* (…) more odds… and when we multiply odd * odd the result is always odd. So, this is quite interesting… because again it shows that, although the concepts are indeed very simple, if one doesn’t actually go through this stuff… then when the moment comes you will be in deep deed trouble if a question is asked that needs this. Currently I have done only 2 days of study and my capacity of retention is pretty ok… for the first time however after hearing this rule, the first little glance of insecurity has come in, thinking… will I be able to remember all these rules? Now, the key is to not be frustrated. Enjoy, enjoy, that’s the whole point. So I am gonna keep enjoy it. I embrace the challenge. Yes, this is awesome… I learned how to quickly factorize, I leearned how to quickly find number of positive divisors of a number, which is so interesting! then I learned that perfect squares have interesting properties such as always having even number of each prime… which then also means they will always have an odd number of positive divisors! 🙂

One question arises… do all numbers that have odd number of total positive divisors are perfect squares?

Well, it actually seems so? DAMN YES! It seems YES! and this is actually a rule that I prefer more than the given one “If a number has han odd number of positive divisors, it is a perfect square for sure” and “All numbers that are NON perfect squares, have an EVEN number of positive divisors”. Ok this is quite exciting actually… I checked the previoys notes where I had factorized number yesterday and yes… it is a great super interesting pattern that was there but I had not realized. Awesome.

Ok another question:

 

So here having properly studied and understood all concepts before, this question becomes VERY EASY. I directly went to factorize 2520 and found out that I was missing 2*5*7 to pair up each factor to make sure I had  a combiantion of even factors each time.

Now, interestingly, it took me 1m 20sec to do it… so even in a situation where a question has become pretty straightfrward, I still take +1m to solve, which is scary and shows how hard it is to respond each question quickly and correctly without mistakes. Now, I truly hope that I will get quicker as time goes by. You will see my progress if you keep following this thread.

Divisor Rules for INTEGERS (important that all num are integers).

Rule 1: For k,M,N as integers, if k is a divisor of N, then k is a divisor of NM

Now even before the video explains why is this the case, I am gonna try answer it myself. I mean… if k is a divisor of N, this means k is a factor of N, this means N, in its components will have k hidden somewhere. Then obviously, if I multiply N by any other integer… k will STILL be inside its components. This makes more sense to me than the rule itself. Basically if a number N has k as one of its divisors, then multiplying N by another integer M, will still have k inside its components no matter what. This seems obvious, I don’t know why they don’t frame things like this instead of complicating it.

Rule 2: given j,k and N are integers, if jk is a divisor of N, then j is a divisor of N and k is a divisor of N

Now, this again same thing. Obviously, if N has j*k in its components… this means N has j in its components, and it means N has k in its components. Pretty clear. So basically if 10 is a divisor of 100, then 5 is a divisor of 100 and 2 is also a divisor of 100. I like to see that, having understood the fundamentals in depth, I am able to really get this pretty quickly.

Rule 3: if k is not a divisor of N then jk is not a divisor of N.

Ok, this wasn’t AS obvious as before but if we think about it it also should make sense. Basically what we are saying is… (btw, it just made total sense, actually is not hard at all), that k is NOT a component of N. Then obviously we know that jk is not a component of N, because for sure k at least is not welcomed in N XD! And the interesting thing is that it doesn’t matter what j is. So for example 3 is not a divisor of 100 (2*2*5*5), hence no matter what you multiply 3 by, it will never be welcomed in N(100). How freaking picky these numbers right? That 3 must be feeling rejected. 🙁 feel sad for it.

Rule 4: if k is a divisor of both N and M then K is a divisor of N+M (and N-M and M-N).

Damn, this sarts getting complex. Ok thought for 2 min and got why even if it is not explained in the video.

Imagine N = 35 (7*5) and M = 21 (7*3). Now, as you can see, 7 is a divisor of 35 and 7 is a divisor of 21, hence the rule says 7 will also be a divisor of (35+21) and (35-21) and (21-35). Why?

Because 35 + 21 = (7*5) + (7*3) which = 7*(5+3), so you can see… no matter what is in the brackets, as both numbers shared 7 as a divisor, the 7 will always be there. So even if it is 7(5-3) or 7(3-5) or 7(3+5), the 7 will always be there.

ALARM SOUNDED 2:30 hours of work till here. I am meeting someone later but wanna do more :D! So I put another alarm of 1:30 hours.

Now, conceptually, what it is important to understand is that, if 2 numbers share the same prime as a factor (so that k number is both a divisor of N and M), then that number will also be a divisor of the addition or substraction of those numbers. Done.

Rule 5: if k is  divisor of N but k is NOT a divisor of M, then k is NOT a divisor of N+M or (N-M) or (M-N)

I was kind of expecting rule 5 to be like this XD! Interesting… hmm… in same previous example, 5 is a fivisor of 35, but not a divisor of 21, so this means that when we do this: 7(5-3) or 7(3-5) or 7(3+5) you can see that… for sure 5 won’t NECESSARILY be a divisor of M… now the rule says it NEVER is. Why… well it is a bit harder to understand but for example:

7(5-3) you can see that if I want to make the brackets end up being a multiple of 5, then that prime factor of M (which in this case is 3) should be either 10, 15, 20, or -10 etc etc… so what I am saying is… if I want the brackets to end up being a multiple of the prime factor of N, then that prime factor of M has to ALSO be a multiple of 5, which is impossible, because we said specifically that 5 was NOT a factor (or a component) of M. That’s why.

Now I understand these concepts but I am a bit scared regarding how to implement these quickly enough. But it’s ok, I am confident. Knowing these concepts will come back once I start actually tackling the problem sets. Let’s see how that goes once I start. But first I wanna study the whole topic.

A great example of why it is actually hard to implement this:

Basically seems a simple question? Took me 1m 30s to conclude that 1) is already enough to know that yes, it is divisible. But then for 2) I wasn’t fully sure if it was sufficient on its own, or if gave any info useful for us. I kind of assumed that if x+y is divisible by 6 then x and y would also be but WRONG. VERY VERY careful.

We said that if k is a divisor of x and k is a divisor of y, then k will be a divisor of (x+y). We DIDN´t say that if k is a divisor of x+y then it must be a divisor of each. For example 4 + 2 = 6. Is 6 a divisor of 6? yes. Is 6 a divisor of 4? NO. and of 2 ? NO. Hence this statement does not really help us at all, it just confuses us. If in doubt, better to try find a case that disproves the statement like above (if one has time). Will keep this in mind. I made a mistake… I shouldn’t make mistakes here. I should be sharper. I will be sharper.

Found also very interesting these different frames of the same question:

You can see they all mean the same. I think I am almost at the point where they all sounds the same to me… maybe 80% there. Need to encounter and do more problems to really interiorize it.

The most counterintuitive one to me is the one “Is there some integer k such that 6k = x”? But then I thought… obviously if X is built by doing 6* (something), then obviously what this means is that 6 must be a factor of X, or a divisor of X.

Greatest common divisors

Now this should be easy. Basically we factorize two numbers and pick all common factors and multiply them together to find the greatest common divisor.

What if there is no prime factor that is common between 2 numbers? Then the greatest common divisor is 1. Again this is something basic but could cause confusion if not studied.

Here an example:

Now this took me 28s to solve. Pretty quickly. As it should be. However as it was so “easy” it did cause a bit of uncertainty in me… did I miss something? etc. Careful with this. I wonder in this situation if it is better to take 20s more to double check, or just move on. We will see as I do more exercises.

Done with this topic. At the end of the day all falls into factorizing and understanding that well. This is just asking what is the max common out of those.

Least common multiple

The least common multiple of integers x and y is the smallest positive integer that has x and y as its divisors.

Again not sure why complicating. This just means that if we have two numbers x and y, the LCM is the smallest number divisible by both, which basically means the smallest number that contains all “components” of both x and y. So if x (2*2*3*7) and y (3*5*13), then what is the smallest number that is contained in both? Well (2*2*3*5**7*13), you can see that both numbers are contained in there. Easy

Interesting to understand “Why” is this important. Well video shows one good example. When we add fractions, we are actually doing LCM under the lines without realizing!

Relation between GCD and LCM

Actually before the video explained it I stopped it and deduced it myself. You can see that if you have 2 numbers for instance 18 (2*3*3) and 42 (2*3*7) then one can see that : GCD * (M/GCD)*(N/GCD) = LCM. This if you simplifie it is equal to : LCM * GCD = L*M

Even and odd numbers and their relationships

Every integer is either even or odd.

1. Even and odd numbers include negative values
2. o is even
3. Even n are divisible by 2 // Even numbers can be expressed as 2k where k is an integer // Prime factorization of even will include at least one 2
4. Odd numbers are NOT divisible by 2 (this is why in a multiplication whenever an even num is involved, the final num is even always).
5. Odd numbers can be expressed as 2k + 1 where k is an integer
6. Odd (+ or -) odd = even  (Makes sense, basically means I am moving on the line of numbers in odd jumps, which will land me on an opposite kind of number)
7. Odd (+ or – ) even = odd (Makes sense, basically means I am moving on the line of numbers in pairs which will of course land me on another odd)
8. Actually although there are rules for 6 and 7, best way is to understand that if I jump odd n of positions  then the nature of my num changes. If I jump even n of positions the nature of the number remains.
9. In multiplication we already said, that as soon as even comes in, will be always even. Only when odd x odd then result is odd.
10. Divison is trickier. EVEN / EVEN can be non integer, even or odd (Makes sense, as it is easy to be EVEN and u can be it even if u have odd components inside)
11. ODD / EVEN – CANNOT be an integer (Makes sense, u can’t split a num with no 2 in its component by 2)
12. EVEN / ODD – not integer, or even (again, it can certainly never be odd, cause that pesky 2 present on numerator will always be there. so it will either be even (when numerator contains some odd that cancel out with the denominator ones) or it will be NON INTEGER, whenever there is no way to fully cancel out those odd nums on denominator.
13. ODD / ODD – basically can be either odd or non integer… will never be even because there is no 2’s to work with

Now let’s see a problem:

 

This is pretty hard. It took me 3m 17s  to get to a wrong answer… which was just because I overlooked something but the approach was right. I guess with practice I’ll improve but I need to be EXTRA careful.

So first important thing is to understand what the sum of 2 componenst can give ODD. Thinking a bit one discovers that only if One component is ODD and one EVEN, then this happens. So on first formula, let’s assume y is the odd component, in that case xy should be even, which can happen if x is even ONLY. If we check other way around, let’s assume Y is even, then xy has to be odd, which is impossible because Y is even and is already there. Hence we are sure X is even. SUFFICIENT.

On second empression, we see a substraction of two components. We know that if result is odd it means that one component is odd other is even. lets assume 6x is odd, well it can’t , so let’s assume 6x is the even then this means x can either be odd or even, and 3y is odd which means y is odd… hence we don’t know if X is actually odd or even from this expression. INSUFFICIENT.

ALARM sounded 1:30 hours went by. Will do 45 min more.

So here the main issue I had was that when considering xy even and y odd, I failed to realize that x MUST be of course even for xy to be even TAKING INTO ACCOUNT Y is odd. That last bit was the main issue. I forgot I knew that Y was odd in that case. Need to be a bit more careful next time.

Very interestingly in the video this is solved in another way:

For expression 1 we turn into y(1+x) is odd. Which means Y is odd and 1+x is odd hence X is even for sure.

But interestingly there is an even further technique that can be used and that I must test for sure each time to understand what is best. This technique is more of a “brute force” one but might actually be the best:

Now let me try this method see how long it takes…. ok it took me 3m 19s but the method gives a lot of certainty and peace. I will have to rehearse more but I think I will end up using this method. Next time I encounter similar problem will use it and see.

Ok this is pretty awesome. Consider this exercise:

I used the table and got an answer 1m 19s, not bad at all. This is very good news, as it really helps simplify the questions. Just need to be careful to understand instead of blindly trust. I like the table technique a lot because in questions where you need to determine what is sufficient or not etc… then you know you are taking ALL cases in consideration. Also, in this case calculations were so easy. Remember we chose 1 for odd and 2 for even. This caused a lot of 0’s in the equation making the calculations super quick. I really like to know that 0 is even and can be used without fear in these kinds of problems. Great.

Properties of consecutive integers

Rule 1: In general every nth number is divisible by n

This means every 6th number is divisible by 6 for instance. Makes sense so far. Now if we have 2 consecutive numbers, this means that one of them MUST be divisible by 2! if we have 3 consecutive we know one of them MUST be divisible by 3 etc.

Rule 2: If we have n consecutive numbers, one of those numbers MUST be divisible by n.

Now here an important issue I just faced. The question is… What conclusions can one take from this expression:

When I saw it, I thought ok let’s simplify by doing n(n^2 – 1). Now, just by doing that I was missing a MASSIVE piece of information which is that (n^2 – 1) can actually be simplified even more to (n – 1) * (n + 1). This is of MASSIVE importance because if I had known, then I’d know we are talking about the product of 3 consecutive numbers…

This is something a bit tricky because we jump into Algebra which is a topic I have not yet revised, and that is very important.

Difference of squares rule

The rule being used here is the “Difference of squares rule”, that says that (a^2 – b ^2) = (a – b) (a + b).  In this case I think because I already learned this in high school long time back, this was stored somewhere in my brain… so now that I see it I will be more careful next time for sure.

Remainders and terminology

Ok so when we divide for instance 15 / 2, we get 7 and 1 remainder. Quocient is the answer to a division problem. Divisor is the number we Divide by, and Dividend sounds like “dividiendo” in spanish which means the number you are diving (in this case 15). Fine, this is clear.

Rule 1: The remainder is always greater than 0 and lower than the divisor (This is super obvious)

Alarm soundedn – 45 minutes passed, actually more, about 1:00 hour I’d say.

I’m back.

Rule2: If N is divided by D, and remainder is R, then possible values of N are: R, D + R, 2D + R etc. 

This is a bit counterintuitive but seems important. Let’s say D is 7 and R is 2, then how can R be  a possible value of N? Well, because 2 / 7  = 0 with remainder 2! Careful with this.

The exercise above is damn easy but took me still 1m 11s… just because I wasn ‘t fully confident. But from division rules we know that if for example 10/3 = 3 and 1 remainder… 3*3 + 1 must equal 10. That’s all the equarion is saying.

I think this concept should be totally fine. If later I encounter a hard question in this regard… then I’ll get worried, but overall seems pretty easy.

Another question:

Again super easy hmm… I guess these are the kinds of super easy questions that come at the beginning of the test! Good to know they are not all super hard.

Took me 1m 9s.

Now another interesting question:

This one caught me a bit by surprise to be honest. Took me 2m 11 s to solve which is way too much.

First I tried looking at the equations but got a bit confused, so then changed strategy to just substitute numbers. I tried first with 3, but then when seeing it calculated to 0 I got confused, as I was not sure if 0 is divisible by 6… which it is. (now I learned). Once u go to option D it gives 3, and 3 is NOT divisible by 6. Anyway as I was confused with using 3 which would have been the easier… I chose 15 instead, which gave me the correct answer. D. Clearly its getting late and I am a bit more tired, but glad I got the righr answer anyway.

The other option is to understand that N = x*12 + 3 . (where x is an integer) You can substitute this in each expression, for example on the first it gives 12x. This can be expressed as 6(2x) so yes, it is divisible by 6. On the second expression we get 12x + 18 which again can be expressed as 6(2x+3), which again, it IS divisible by 6 etc etc. 🙂 Proably this method is best, but if a bit blocked, sometimes substituting numbers is best policy.

End of Integer Properties studies

Here I reach the end of the integer properties section of the videos which are being great. I like it so much that I signed up for the free program and will follow it as a foundation: /www.gmatprepnow.com

Overall strategies

Now that I have decided to follow this program first as I feel it is extremely useful and comprehensive, I will start from the beginning. First there is a section about overall strategy which I won’t go into detail unless I encounter problems or truly insightful info. Every problem is a chance to solve and measure so I will take them all. Like this one:

Took me 56s to solve, super easy.

Next one:

Again I think these are just very basic examples to show the kind of questions one can encounter:

And that was also pretty easy. Solved in 59s.  I can totally see Data sufficiency becoming really hard though. Will need to be very careful.

Although Verbal part is absolutely key no doubt, I won’t focus on verbal part in this blog, mainly because overall I am stronger in verbal and I will study that using other means (Like an app ready4GMAT, and then of course videos etc… but won’t coment here). My goal with this is to get all math questions right. Remember the philosophy… if I understand all root concepts well, there is no reason why I shouldn’t answer each and every question right XD! Let’s see what happens.

Alarm sounds 1 hour has passed. Put 45 more minutes.

Timings

This table for timings is suggested. I am not sure if it is the one I will use or how I will manage my time but I wanna keep it for reference here in case I need to go back to it later:

 

In one way or another for sure I must have a strategy to manage my time. We will go into that after I take some full length practice test and see how I manage.

Cancelling score?

I just learned you can cancel your score (which btw you will be able to see right at the end of the exam), and schools wouldn’t know! This is massive really… makes it so much easier if you have enough money to know u can do 2 attempts for example. If in a week I feel confident enough, I might book a test for 3 weeks after now or so… just to see what my true level is… let’s see. Otherwise I plan to take it in Feb or March or so.

Math strategy

This is something I should be good at, but in the question below I performed poorly:

What I did was do the multiplication. Unfortunately I actually forgot how to multiply decimals :S, so I multiplied by 4 first, and then by 15 (result then divided by 100), to get answer d in 1m 2s.

The stupid thing is that… given the answer choices given, I could have estimated D as being the answer much much quicker.

This is another interesting example:

This one I solved better by estimating. So 50% is 412 more or less, and add roughly 50 more so 460 approx which is D. Solved in 15s. But it would have been even more smarter to realize that 50% of 825 is 412,5 which already means D is correct answer FOR SURE.

Now consider this question:

This one I solved in 1m 24s. For some reason this brought a story to my head, something I read years back about a very famous mathematician who got asked this in high school, and found a method to solve it quickly. First u need to determine how many numbers in that range. I don’t remember so I just wrote 1,2,3, and if I do 3-1 then = 2, but actually there are 3 num in that range, so I need to add 1 at the end. Hence 53-12 = 41, so there are 42 numbers in that range, so 2 equally split halfs of 21 numbers.

The smart thing to realize is that if I add the first and last num of the series, it gives me same number than if I add the second and the second last etc… in this case gives 65. So I know that the answer must then be 21*65 = E.

Also I checked answers quickly and saw they are super close together so no estimation could be made.

I am happy to see that even if I forgot everything, still things I might have learnt years back are somehow coming back.

Technique to divide by 5 quickly.

Well, this is handy to know just in case, but basically to divide a number by 5 quickly I can always double the number and then divide by 10 :). Handy technique maybe.

Alarm sounds Ill put 30 min more.

Consider this:

Now again, this I really got blocked and probably solved in 4min or so. Big issues here. First of all I was blocked by the root as i forgot totally how to calculate roots and also how to manipulate them.

What I did was to divide the numebrs into their factors, which helped but eventually missed the whole point. The interesting thing is to realize that both left and right expressions, contain 9 inside of them. So I can re-write them as 25*9 + 9*24. This I could have easily seen just by looking at the factors but I wasn’t sharp enough. Once one knows that then one can do 9*49, which both obviously are perfect squares that I can do root(9) * root(49) hence 3 * 7 = 21

Summary of GMAT Day 2 Study

Conclusion

I did 7 SOLID hours of study today. Hmm… Interesting I thought it had been longer. I should optimize my day better and try to do 9 or so. Let’s see.

I went a lot in depth on proeprties of integers, primes, divisions and some strategy and overall techniques… now I have slight fear I might forget stuff. Let’s not have that fear, let’s enjoy, remember the purpose is to enjoy enjoy, have fun, understand the root concepts, solve problems and crack it. I’ll try to keep enjoying as much as possible.

Big TODO’s

1. Learn how to multiply and divide by brute force using decimals too just in case
2. Learn how roots work and how to solve a root by brute force (maybe)
3. Although on Day 1 I did combinations and permutations, by no means that is mastered and need to come back
4. Although I have done a lot of in depth study on Properties of Integers, I have not done exercises yet, and should probably start doing them asap to test. Need to understand where I can find exercises of this particular kind. If hard then I’ll aim to complete whole Math review ASAP and then I’ll do holstic tests.
5. My main strategy is gonna be following the program diligently and IN DEPTH. I know I might forget something by the end… but it is then that I will start doing a lot of problems, and then going back to each section whenever I feel I have problems. So far this is what I am gonna do.