GMAT Quant Syllabus 2021-2022
Posted on
22
Jul 2021

GMAT Quant Syllabus 2021-2022

Author: Apex GMAT
Contributor: Altea Sollulari
Date: 22 July, 2021

We know what you’re thinking: math is a scary subject and not everyone can excel at it. And now with the GMAT the stakes are much higher, especially because there is a whole section dedicated to math that you need to prepare for in order to guarantee a good score. There is good news though, the GMAT is not actually testing your math skills, but rather your creative problem solving skills through math questions. Furthermore, the GMAT only requires that you have sound knowledge of high school level mathematics. So, you just need to practice your fundamentals and learn how to use them to solve specific GMAT problems and find solution paths that work to your advantage. 

The Quantitative Reasoning section on the GMAT contains a total of 31 questions, and you are given 62 minutes to complete all of them. This gives you just 2 minutes to solve each question, so in most cases, the regular way of solving math equations that you were taught in high school will not cut it. So finding the optimal problem solving process for each question type is going to be pivotal to your success in this section. This can seem a daunting start, so our expert Apex GMAT instructors recommend that you start your quant section prep with a review of the types of GMAT questions asked in the test and math fundamentals if you have not been using high school math in your day to day life. 

What types of questions will you find in the GMAT quant?

There are 2 main types of questions you should look out for when preparing to take the GMAT exam:

Data Sufficiency Questions

For this type of GMAT question, you don’t generally need to do calculations. However, you will have to determine whether the information that is provided to you is sufficient to answer the question. These questions aim to evaluate your critical thinking skills. 

They generally contain a question, 2 statements, and 5 answer choices that are the same in all GMAT data sufficiency questions.

Here’s an example of a number theory data sufficiency problem video, where Mike explains the best way to go about solving such a question.

Problem Solving Questions

This question type is pretty self-explanatory: you’ll have to solve the question and come up with a solution. However, you’ll be given 5 answer choices to choose from. Generally, the majority of questions in the quant section of the GMAT will be problem-solving questions as they clearly show your abilities to use mathematical concepts to solve problems.

Make sure to check out this video where Mike shows you how to solve a Probability question.

The main concepts you should focus on

The one thing that you need to keep in mind when starting your GMAT prep is the level of math you need to know before going in for the Quant section. All you’ll need to master is high-school level math. That being said, once you have revised and mastered these math fundamentals, your final step is learning how to apply this knowledge to actual GMAT problems and you should be good to go. This is the more challenging side of things but doing this helps you tackle all the other problem areas you may be facing such as time management, confidence levels, and test anxiety. 

Here are the 4 main groups of questions on the quant section of the GMAT and the concepts that you should focus on for each:

Algebra

  • Algebraic expressions
  • Equations
  • Functions
  • Polynomials
  • Permutations and combinations
  • Inequalities
  • Exponents

Geometry

  • Lines
  • Angles
  • Triangles
  • Circles
  • Polygons
  • Surface area
  • Volume
  • Coordinate geometry

Word problems

  • Profit
  • Sets
  • Rate
  • Interest
  • Percentage
  • Ratio
  • Mixtures

Check out this Profit and Loss question.

Arithmetic

  • Number theory
  • Percentages
  • Basic statistics
  • Power and root
  • Integer properties
  • Decimals
  • Fractions
  • Probability
  • Real numbers

Make sure to try your hand at this GMAT probability problem.

5 tips to improve your GMAT quant skills?

  1. Master the fundamentals! This is your first step towards acing this section of the GMAT. As this section only contains math that you have already studied thoroughly in high-school, you’ll only need to revise what you have already learned and you’ll be ready to start practicing some real GMAT problems. 
  2. Practice time management! This is a crucial step as every single question is timed and you won’t get more than 2 minutes to spend on each question. That is why you should start timing yourself early on in your GMAT prep, so you get used to the time pressure. 
  3. Know the question types! This is something that you will learn once you get enough practice with some actual GMAT questions. That way, you’ll be able to easily recognize different question types and you’ll be able to use your preferred solution path without losing time.
  4. Memorize the answer choices for the data sufficiency questions! These answers are always the same and their order never changes. Memorizing them will help you save precious time that you can spend elsewhere. To help you better memorize them, we are sharing an easier and less wordy way to think of them:
  5. Make use of your scrap paper! There is a reason why you’re provided with scrap paper, so make sure to take advantage of it. You will definitely need it to take notes and make calculations, especially for the problem-solving questions that you will come across in this GMAT question.
  • Only statement 1
  • Only statement 2
  • Both statements together
  • Either statement
  • Neither statement
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Posted on
21
Jul 2021

GMAT 3D Geometry Problem – GMAT Math – Quant Section

GMAT 3D Geometry Problem 

In this problem we’re going to take a look at 3D objects and in particular a special problem type on the GMAT that measures the longest distance within a three-dimensional object. Typically, they give you rectangular solids, but they can also give you cylinders and other such objects. The key thing to remember about problems like this one is that effectively we’re stacking Pythagorean theorems to solve it – we’re finding triangles and then triangles within triangles that define the longest distance.

This type of problem is testing your spatial skills and a graphic or visual aid is often helpful though strictly not necessary. Let’s take a look at how to solve this problem and because it’s testing these skills the approach is generally mathematical that is there is some processing because it’s secondary to what they’re actually testing.

gmat 3d geometry question

GMAT 3D Geometry Problem Introduction

So, we have this rectangular solid and it doesn’t matter which way we turn it – the longest distance is going to be between any two opposite corners and you can take that to the bank as a rule: On a rectangular solid the opposite corners will always be the longest distance. Here we don’t have any way to process this central distance so, what we need to do is make a triangle out of it.

Notice that the distance that we’re looking for along with the height of 5 and the hypotenuse of the 10 by 10 base will give us a right triangle. We can apply Pythagoras here if we have the hypotenuse of the base. We’re working backwards from what we need to what we can make rather than building up. Once you’re comfortable with this you can do it in either direction.

Solving the Problem

In this case we’ve got a 10 by 10 base. It’s a 45-45-90 because any square cut in half is a 45-45-90 which means we can immediately engage the identity of times root two. So, 10, 10, 10 root 2. 10 root 2 and 5 makes the two sides. We apply Pythagoras again. Here it’s a little more complicated mathematically and because you’re going in and out of taking square roots and adding and multiplying, you want to be very careful not to make a processing error here.

Careless errors abound particularly when we’re distracted from the math and yet we need to do some processing. So, this is a point where you just want to say “Okay, I’ve got all the pieces, let me make sure I do this right.” 10 root 2 squared is 200 (10 times 10 is 100, root 2 times root 2 is 2, 2 times 100 is 200). 5 squared is 25. Add them together 225. And then take the square root and that’s going to give us our answer. The square root of 225 is one of those numbers we should know. It’s 15, answer choice A.

Okay guys for another 3D and Geometry problem check out GMAT 680 Level Geometry Problem – No Math Needed! We will see you next time.

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7 Daily Practices For GMAT Success - GMAT Guide
Posted on
08
Jul 2021

7 Daily Practices For GMAT Success

By: Apex GMAT
Contributor: Ruzanna Mirzoyan
Date: 8th July 2021

7 Things You Need To Do Daily When Preparing For The GMAT (GMAT Guide)

  1. Visualize success and the value you will get in the end
  2. Review a the GMAT sections
  3. Set a time limit for each day
  4. Do not forget to reward yourself
  5. Forget about the target score only focus on improvement
  6. Give yourself a pep talk 
  7. Evaluate Yourself Honestly

     Achieving a great score on the GMAT exam is not an easy task. The overall preparation process is daunting for a majority of test takers, especially for non-native English speakers. It requires diligent work and a daily checklist that you need to follow. So how do you come up with a plan that works? This article covers seven tips for successful GMAT prep which will guide you throughout the entire process. Even though every individual taking the exam has different expectations, experiences and may be approaching the test in a different way, sticking to a daily routine is an integral part of test success; the most difficult thing is adhering to it, avoiding procrastination and maintaining motivation. Therefore, after learning all the exam basics, such as the timing, the sections, and the preparation materials, it is worth creating a checklist to help keep you on track.

Visualize success and the value you will get in the end

The thought of success can create happiness! Once we attain something that seemed difficult initially, the suspense wears off, and the excitement rapidly grows. By taking time every day to imagine achieving your goal you can stay motivated and on the right path. When we experience happiness our brain releases serotonin, the hormone responsible for happiness. By keeping the picture of accomplishment in our mind, this happiness never fades. Hence, if every day contains even a tiny bit of happiness, even the most complex struggles seem simpler to overcome. Whether the GMAT exam is a struggle or not, happiness and motivation are something that one undoubtedly always lacks. Do your best to look at the bigger picture and think of the steps that will expedite reaching the top.

Review the GMAT exam sections

Whether you have a private GMAT tutor or are studying on your own, be sure to review difficult parts of the overall format of the exam every day before going through your study materials, for example the data sufficiency answer choices. You may do a short quiz on quantitative, verbal, or integrated reasoning to keep pace with timing and question types. You can consider this form of revision as stretching your brain muscles before the main exercise. Doing a simple GMAT quiz each time will make you more cautious about time management and remind you about the type of questions that you may have already mastered in previous study sessions.

Set a study time limit for each day

As it is said, time is the only non-redeemable commodity, so proper allocation is a fundamental key to success. We recommend you have a specific time allocation for GMAT prep each day. That can be some time for weekday preparation and extension on the weekends. Ensure the limit you set for yourself is reasonable because procrastinating one day and doubling the hours the next day does not work out. It does not matter how many months you have on your hands; the significant thing is precise allocation. If you want to get a decent score, you must spend approximately 100-120 hours reviewing the materials and practicing. However, top scorers usually  spend 120+ hours studying. Whether you belong to the former or the latter category, remember that time is the most expensive investment you are making. At the same time keep in mind that your study-life balance should be of utmost importance. 

Do not forget to reward yourself

It is not a secret that the GMAT is burdensome and overwhelming, and preparing for it can be stressful and oftentimes disheartening. Not having small rewards to look forward to can lead to demotivation. Rewards are things that rejuvenate your broken concentration. Try something like the Pomodoro Technique. This technique helps break down time into intervals with short breaks. Instead of breaks, you can think of something ‘non-GMAT related’ that will make you regain focus. For example, by grabbing a quick snack, meditating, or walking around the house or even watching a short YouTube video. Whichever works best for you, make use of it; even brief respites retain your stamina. Finally, never forget about the bigger reward; your final score. 

Forget about the target score, only focus on improvement

GMAT preparation practices do generate plight both in physical and mental states. It is crucial to remind oneself of the improvement phases. We agree that everything you are going through is for the final score. But focusing on the final score too much can frustrate you if you are not making big leaps towards it, which in turn can be counter productive. All successful practices dictate that you should focus on one thing at a time, which improves every day until the exam day. When the exam day comes, you will utilize all the knowledge and effort to get the highest GMAT score possible. Keeping daily track of your improvements relieves some of the burden on your shoulders. Even the tiniest advantage acquired can be a game changer. For instance, finishing each section a minute earlier than before will eventually contribute to achieving more significant results on the exam day, or perfecting a solution path which has you approaching a host of GMAT problems in a more efficient manner. These small wins can be the fuel to keep you going. 

Give yourself a pep talk 

I am sure you receive a lot of support from the people surrounding you. However, self-encouragement is of the utmost importance. Look around, see what others are doing at your age and inspire yourself. Choose wisely between the tradeoffs. Such as choosing to study instead of partying. Giving yourself a daily pep talk will make you more enthusiastic about reaching your objectives. A recent scientific study has shown that talking to yourself dwindles anxiety and stress while boosting performance. This is no less true for GMAT test preparation. Give yourself motivational and instructional pep talks. This method promotes positivity as motivational talks cheer you up and keep up the eagerness to study and strive for more, while a self-instructional talk directs detail-orientation and accentuates what exactly you need to do for that particular day. For example, start every day by loudly stating what should be done for the day. It helps with thinking about the mechanisms of every individual task and visualizing methods to complete them correspondingly. 

Evaluate Yourself Honestly

Of course, you need all the encouragement and self-support to reach your goals, but especially during GMAT exam preparation, you need to be hard on yourself if required. If you need a 650+ GMAT score, you should be aware that it will not be a piece of cake. Give yourself credit for what you are doing right, but also consider aspects of the GMAT problems that you need to elaborate on and master additional skills. The dominant thing is separating the action from the person because you are evaluating your actions and not you as a person; you should not upset yourself but rather detect the triggers of low performance and challenges and make yourself accountable for such actions with a plan to move forward from them successfully. Ultimately, the ability to discern your flaws and work on personal evolution is an inherent quality for capacitating your abilities and aptitudes and pulling it off in life. 

We hope that adding these practical and mindful aspects to your daily preparation will be helpful as when you are preparing for an exam like the GMAT, being in the right mind frame can be as important as doing the quant or verbal practice. Whether you have a GMAT private tutor or not, it is on you to maintain motivation during the entire process. We suggest you develop a GMAT test strategy along with these seven tips to attain greater productivity and manifest superb performance. Make studying for the GMAT a daily habit and success will follow. 

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The basics of GMAT Combinatorics
Posted on
24
Jun 2021

The Basics of GMAT Combinatorics

By: Apex GMAT
Contributor: Svetozara Saykova
Date: 24th June 2021

Combinatorics can seem like one of the most difficult types of questions to come across on the GMAT. Luckily there are not many of them within the exam. Still these questions make up the top level of scoring on the test and therefore it is best if you are well equipped to solve them successfully, especially if you are aiming for a 700+ score. The most important rule to follow when considering this question type is the “Fundamental Counting Principle” also known as the “Counting Rule.” This rule is used to calculate the total number of outcomes given by a probability problem. 

The most basic rule in Combinatorics is “The Fundamental Counting Principle”. It states that for any given situation the number of overall outcomes is equal to the product of the number of each discrete outcome.

Let’s say you have 4 dresses and 3 pairs of shoes, this would mean that you have 3 x 4 = 12 outfits. The Fundamental Counting Principle also applies for more than 2 options. For example, you are at the ice cream shop and you have a variety of 5 flavors, 3 types of cones and 4 choices for toppings. That means you have 5 x 3 x 4 = 60 different combinations of single-scoop ice creams. 

The Fundamental Counting Principle applies only for choices that are independent of one another. Meaning that any option can be paired with any other option and there are no exceptions. Going back to the example, there is no policy against putting sprinkles on strawberry vanilla ice cream because it is superb on its own. If there were, that would mean that this basic principle of Combinatorics would not apply because the combinations (outcomes) are dependent. You could still resort to a reasoning solution path or even a graphical solution path since the numbers are not so high. 

Let’s Level Up a Notch

The next topic in Combinatorics is essential to a proper GMAT prep is  permutations. A permutation is a possible order in which you put a set of objects.

Permutations

There are two subtypes of permutations and they are determined by whether repetition is allowed or not.

  • Permutations with repetition allowed

When there are n options and r number of slots to fill, we have n x n x …. (r times) = nr permutations. In other words, there are n possibilities for the first slot, n possibilities for the second and so on and so forth up until n possibilities for position number r.

The essential mathematical knowledge for these types of questions is that of exponents

To exemplify this let’s take your high school locker. You probably had to memorize a 3 digit combination in order to unlock it. So you have 10 options (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) for 3 available slots. The total number of locker passwords you can have is 103 = 1,000. 

  • Permutation without repetition allowed 

When repetition is restricted in the given GMAT problem, we would have to reduce the number of available choices for each position. 

Let’s take the previous example and add a restriction to the password options – you cannot have repeating numbers in your locker password. Following the “we reduce the options available each time we move to the next slot” rule, we get 10x9x8 = 720 options for a locker combination (or mathematically speaking permutation). 

To be more mathematically precise and derive a formula we use the factorial function (n!). In our case we will take all the possible options 10! for if we had 10 positions available  and divide them by 7!, which are the slots we do not have. 

10! =  10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 

7! =  7 x 6 x 5 x 4 x 3 x 2 x 1 

And when we divide them (7 x 6 x 5 x 4 x 3 x 2 x 1) cancels and we are left with 10 x 9 x 8 = 720. 

Pro tip: Taking problems and deeply examining them by running different scenarios, and changing some of the conditions or numbers is a great way to train for the GMAT. This technique will allow you to not only deeply understand the problem but also the idea behind it, and make you alert for what language and piece of information stands for which particular concept. 

So those are the fundamentals, folks. Learning to recognize whether order matters and whether repetition is allowed is essential when it comes to Combinatorics on the GMAT. Another vital point is that if you end up with an endless equation which confuses you more than helps, remember doing math on the GMAT Quant section is not the most efficient tactic. In fact, most of the time visualizing the data by putting it into a graph or running a scenario following your reasoning are far more efficient solution paths. 

Feeling confident and want to test you GMAT Combinatorics skills? Check out this GMAT problem and try solving it. Let us know how it goes!

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13
May 2021

GMAT Factors Problem – 700 Level GMAT Question

GMAT Factors Problem

Hey guys! Today we’re going to take a look at one of my favorite problems. It’s abstract, it’s oddly phrased and in fact the hardest part for many folks on this problem is simply understanding what’s being asked for. The difficulty is that it’s written in math speak. It’s written in that very abstract, clinical language that if you haven’t studied advanced math might be new to you.

How this breaks down is they’re giving us this product from 1 to 30, which is the same as 30!. 30*29*28 all the way down the line. Or you can build it up 1*2*3*……*29*30.

The Most Difficult Part of The GMAT Problem

And then they’re asking this crazy thing about how many k such that three to the k. What they’re asking here is how many factors of three are embedded in this massive product. That’s the hard part! Figuring out how many there are once you have an algorithm or system for it is fairly straightforward. If we lay out all our numbers from 1 to 30. And we don’t want to sit there and write them all, but just imagine that number line in your head. 1 is not divisible by 3. 2 is not divisible by 3, 3 is. 4 isn’t. 5 isn’t. 6 is. In fact, the only numbers in this product that concern us are those divisible by 3. 3, 6, 9, 12, 15, 18, 21, 24, 27, 30.

Important Notes About Factors

Here it’s important to note that each of these components except the three alone has multiple prime factors. The three is just a three. The six is three and a two. The nine notice has a second factor of three. Three times three is nine and because we’re looking at the prime factors it has two. It’s difficult to get your head around but there are not three factors of three in nine when you’re counting prime factors.

Three factors of three would be 3 by 3 by 3 = 27. So notice that 3 and 6 have a single factor. 9 has a double factor. Every number divisible by 3 has one factor. Those divisible by 9 like 9, 18 and 27 are going to have a second factor and those divisible by 27, that is 3 cubed, are going to have a third factor. If we lay it out like this we see ten numbers have a single factor. Another of those three provide a second bringing us to thirteen. Finally, one has a third bringing us to fourteen. Answer choice: C.

GMAT Problem Form

So let’s take a look at this problem by writing a new one just to reinforce the algorithm. For the number 100 factorial. How many factors of seven are there? So first we ask ourselves out of the 100 numbers which ones even play? 7, 14… 21 so on and so forth. 100 divided by 7 equals 13. So there are 13 numbers divisible by 7 from 1 to 100. Of those how many have more than one factor of 7? Well we know that 7 squared is 49. So only those numbers divisible by 49 have a second factor. 49 and 98. There are none that have three factors of 7 because 7 cubed is 343. If you don’t know it that’s an identity you should know. So here our answer is 13 plus 2 = 15.

Try a few more on your own. This one’s great to do as a problem form and take a look at the links below for other abstract number theory, counting prime type problems as well as a selection of other really fun ones. Thanks for watching guys and we’ll see you soon.

If you enjoyed this GMAT factors problem, here is an additional number theory type problem to try next: Wedding Guest Problem.

 

 

 

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Posted on
14
Apr 2021

GMAT Percentage Problem – Unemployment Rate -Multiple Solution Paths

GMAT Percentage Problems

Hey guys, GMAT Percentage problem/s are commonplace on the GMAT and today we’re going to take a look at one that is straightforward but could very easily get you caught up with the math. In this problem, notice that there’s the word “approximately.” That always means there’s an Estimation Solution Path. We’ll take a look at that first but then we’re going to look at a Scenario Solution Path, which for many people is a lot more natural. In addition to seeing that word approximately you can see that there’s this massive spread within the answer choices. Once again pushing us towards an Estimation Solution Path.

Estimation Solution Path

So let’s dive in: The unemployment rate is dropping from 16% to 9% and your quick synthesis there should be: okay it’s being cut about in half or a little less than half. And monitoring that directionality is important. Additionally, the number of workers is increasing. So we have lower unemployment but a greater number of workers. So we have two things, two forces working against one another. If the number of workers were remaining equal then our answer would be about a 50% decrease or just under a 50% decrease, so like 45% or something like that. But because we’re increasing the number of workers, our decrease in unemployment is lower. That is we have more workers, so we have a larger number of unemployed so we’re not losing as many actual unemployed people and therefore our answer is B: 30% decrease.

Scenario Solution Path

If we want to take a look at this via Scenario, we can always throw up an easy number like 100. We begin with 100 workers and 16% are unemployed so 16 are unemployed. Our workers go from 100 to 120. 9% of 120 is 9 plus 0.9 plus 0.9 = 10.8% or 11%. What’s the percentage decrease from 16 to 11? Well it’s not 50, that’s too big. It’s not 15, that’s too small. It’s about 30 and the math will bear us out there.

So thanks for watching guys! Check out the links below for other GMAT percentage problem/s and we look forward to seeing you again real real soon.

Another GMAT percentage problem

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GMAT Prime Factorization Article
Posted on
01
Apr 2021

GMAT Prime Factorization (Anatomy of a Problem)

By: Rich Zwelling (Apex GMAT Instructor)
Date: 1st April 2021

First, if you’d prefer to go straight to the explanation for the solution to the problem given in the last post, continue to the end of this post. But we’ll start with the following Official Guide GMAT problem as a way to talk about GMAT Prime Factorization. Give the problem a shot, if you can:

How many prime numbers between 1 and 100 are factors of 7,150?

A) One
B) Two
C) Three
D) Four
E) Five

One of the things you’ll notice is the linguistic setup of the problem, which is designed to confuse you immediately (a common theme on GMAT problems). They get you panicking right away with mention of a large range (between 1 and 100), and then they compound your frustration by giving you a rather large value (7,150). 

Don’t let that convince you that you can’t do the problem. Because remember, the GMAT is not interested in large calculations, memorization involving large numbers, or weird arcana. Chances are, if you find yourself thinking about a complicated way to do a problem, you’re taking the wrong approach, and there’s a simpler way.

Try to pick out the most operative signal words, which let you know how to address the problem. We are dealing with prime numbers and also the topic of finding factors. The language of the problem may make you nervous, thinking that we must consider a slew of prime numbers up to 100. But the only primes we are really interested in are those that are actually factors of 7,150. 

So let’s focus our attention there. And we can do so with a prime factor tree. Does this bring back memories? 

Now, in the case of 7,150, we don’t have to break it down into prime numbers immediately. Split the number up into factors that are easy to recognize. In this case, the number ends in a zero, which means it is a multiple of 10, so we can start our tree like this:

prime factorization on the GMATNotice that the advantage here is two-fold: It’s easier to divide by 10 and the two resulting numbers are both much more manageable. 

Splitting up 10 into it’s prime factorization is straightforward enough (2 and 5). However, how do we approach 715? Well, it’s since it ends in a 5, we know it must be divisible by 5. At that point, you could divide 715 by 5 using long or short division… 

…or you could get sneaky and use a NARRATIVE approach with nearby multiples:

750 is nearby, and since 75/5 = 15, that must mean that 750/5 = 150. Now, 750 is 35 greater than 715. And since 35/5 = 7, that means that 715 is seven multiples of 5 away from 750. So we can take the 150, subtract 7, and get 143

Mathematically, you can also see this as: 

715/5 = (750-35)/5 = 750/5 – 35/5 = 150 – 7 = 143

So as stands, here’s our GMAT prime factorization:

prime factorization GMAT articleNow, there’s just the 143 to deal with, and this is where things get a bit interesting. There are divisibility rules that help make factoring easier, but an alternative you can always use is finding nearby multiples of the factor in question. 

For example, is 143 divisible by 3? There is a rule for divisibility by 3, but you could also compare 143 against 150. 150 is a multiple of 3, and 143 is a distance of 7 away. 7 is not a multiple of 3, and therefore 143 is not a multiple of 3.

This rule applies for any factor, not just 3. 

Now we can test the other prime numbers. (Don’t test 4 and 6, for example. We know 143 is not even, so it’s not divisible by 2. And if it’s not divisible by 2, it can’t be divisible by 4. Likewise, it’s not divisible by 3, so it can’t be divisible by 6, which is a multiple of 3.)

143 is not divisible by 5, since it doesn’t end in a 5 or 0. It’s not divisible by 7, since 140 is divisible by 7, and 143 is only 3 away. 

What about 11? Here you have two options: 

  1. Think of 143 as 110+33, which is 11*10 + 11*3 → 11*(10+3) → 11*13
  2. If you know your perfect squares well, you could think of 143 as 121+22 

→ 11*11 + 11*2 → 11*(11+2) → 11*13

Either way, you should arrive at the same prime factorization:

GMAT prime factorization QuestionNotice that I’ve marked all prime numbers in blue. This result shouldn’t be a surprise, because notice that everything comes relatively clean: there are only a few prime numbers, they are relatively small, and there is just one slight complication in solving the problem (the factorization of 143). 

So what is the answer? Be very careful that you don’t do all the hard work and falter at the last second. There are five ends to branches in the above diagram, which could lead you prematurely to pick answer choice E. But two of these branches have the same number (5). There are actually only four distinct primes (2, 5, 11, 13). The correct answer is D.

And again, notice that the range given in the question stem (1 to 100) is really a linguistic distraction to throw you off track. We don’t even go beyond 13. 

Next time, we’ll talk about the fascinating topic of twin primes and how they connect to divisibility.

Find other GMAT Number Theory topics here:
Odds and Ends (…or Evens)
Consecutive Integers (plus more on Odds and Evens)
Consecutive Integers and Data Sufficiency (Avoiding Algebra)
GMAT Prime Factorization (Anatomy of a Problem)
A Primer on Primes

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GMAT Prime numbers article with questions
Posted on
29
Mar 2021

A Primer on Primes

By: Rich Zwelling (Apex GMAT Instructor)
Date: 30th March 2021

As I said in my previous post, GMAT Prime Numbers are my favorite topic. This is because not only are they inherently interesting mathematically but they show up in unexpected circumstances on GMAT problems, even when the term “prime” is not explicitly mentioned.

But before we get to that, I thought it would help to review a basic definition:

If you’ve gone through school, you’ve likely heard the definition of a prime as “any number that can be divided only by 1 and itself.” Or put differently, “any number that has only 1 and itself as factors.”  For example, 3 is a prime number, because 1 and 3 are the only numbers that are factors of 3.

However, there is something slightly problematic here. I always then ask my students: “Okay, well then, is 1 prime? 1 is divisible by only 1 and itself.” Many people are under the misconception that 1 is a prime number, but in truth 1 is not prime

There is a better way to think about prime number definitionally:

*A prime number is any number that has EXACTLY TWO FACTORS*

By that definition, 1 is not prime, as it has only one factor

But then, what is the smallest prime number? Prime numbers are also by definition always positive, so we need not worry about negative numbers. It’s tempting to then consider 3, but don’t overlook 2. 

Even though 2 is even, it has exactly two factors, namely 1 and 2, and it is therefore prime. It is also the only even prime number. Take a moment to think critically about why that is before reading the next paragraph…

Any other even number must have more than two factors, because apart from 1 and the number itself, 2 must also be a factor. For example, the number 4 will have 1 and 4 as factors, of course, but it will also have 2, since it is even. No even number besides 2, therefore, will have exactly two factors. 

Another way to read this, then, is that every prime number other than 2 is odd

You can see already how prime numbers feed into other number properties so readily, and we’ll talk much more about that going forward. But another question people often ask is about memorization: do I have to memorize a certain number of prime values? 

It’s good to know up to a certain value. but unnecessary to go beyond that into conspicuously larger numbers, because the GMAT as a test is less interested in your ability to memorize large and weird primes and more interested in your reasoning skills and your ability to draw conclusions about novel problems on the fly. If you know the following, you should be set (with some optional values thrown in at the end):

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, (41, 43)

Thankfully, you’ll notice the list is actually pretty manageable. 

(And an interesting note that many people forget that 27 is actually not prime. But don’t beat yourself up if this happens to you: Terence Tao, one of the world’s leading mathematicians and an expert on prime numbers, actually slipped briefly on national television once and said 27 was prime before catching himself. And he’s one of the best in the world. So even the best of the best make these mistakes.)

Now, here’s an Official Guide problem that takes the basics of Prime Numbers and forces you to do a little reasoning. As usual, give it shot before reading the explanation:

The product of all the prime numbers less than 20 is closest to which of the following powers of 10 ?

A) 109
B) 108
C) 107
D) 106
E) 105

Explanation

For this one, you have a little hint going in, as we’ve provided you with the necessary list of primes you’ll use to find the product.

And the language given (“closest to”) is a huge hint that you can estimate:

2*3*5*7*11*13*17*19 ~= ??

Since powers of 10 are involved, let’s try to group the numbers to get 10s as much as possible. The following is just one of many ways you could do this, but the universal easiest place to start is the 2 and the 5, so let’s multiply those. We’ll mark numbers we’ve accounted for in red:

(2*5)*3*7*11*13*17*19 ~= ??

10*3*7*11*13*17*19 ~= ??

Next, we can look at the 19 and label it as roughly 20, or 2*10:

10*3*7*11*13*17*19 ~= ??

10*3*7*11*13*17*20 ~= ??

10*3*7*11*13*17*2*10 ~= ??

We could also take the 11 and estimate it as another 10:

10*3*7*11*13*17*2*10 ~= ??

10*3*7*10*13*17*2*10 ~= ??

At this point, we should be able to eyeball this. Remember, it’s estimation. We may not know 17*3 and 13*7 offhand. But we know that they’re both around or less than 100 or 102. And a look at the answer choices lets us know that each answer is a factor of 10 apart, so the range is huge. (In other words, estimation error is not likely to play a factor.)

So it’s not unreasonable in the context of this problem to label those remaining products as two values of 102:

10*3*7*10*13*17*2*10 ~= ??

10*(102)*10*(102)*2*10 ~= ??

And at this point, the 2 is negligible, since that won’t be enough to raise the entire number to a higher power of 10. What do we have left?

101*(102)*101*(102)*101 ~= 107 

The correct answer is C. 

Next time, we’ll get into Prime Factorizations, which you can do with any positive integer.

Find other GMAT Number Theory topics here:
Odds and Ends (…or Evens)
Consecutive Integers (plus more on Odds and Evens)
Consecutive Integers and Data Sufficiency (Avoiding Algebra)
GMAT Prime Factorization (Anatomy of a Problem)
A Primer on Primes

 

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Consecutive Integers and Data Sufficiency (Avoiding Algebra) Article
Posted on
25
Mar 2021

Consecutive Integers and Data Sufficiency (Avoiding Algebra)

By: Rich Zwelling (Apex GMAT Instructor)
Date: 25 March 2021

Last time, we left off with the following GMAT Official Guide problem, which tackles the Number Theory property of consecutive integers. Try the problem out, if you haven’t already, then we’ll get into the explanation:

The sum of 4 different odd integers is 64. What is the value of the greatest of these integers?
(1) The integers are consecutive odd numbers
(2) Of these integers, the greatest is 6 more than the least.

Explanation (NARRATIVE or GRAPHIC APPROACHES):

Remember that we talked about avoiding algebra if possible, and instead taking a narrative approach or graphic approach if possible. By that we meant to look at the relationships between the numbers and think critically about them, rather than simply defaulting to mechanically setting up equations.

(This is especially helpful on GMAT Data Sufficiency questions, on which you are more interested in the ability to solve than in actually solving. In this case, once you’ve determined that it’s possible to determine the greatest of the four integers, you don’t have to actually figure out what that integer is. You know you have sufficiency.)

Statement (1) tells us that the integers are consecutive odd numbers. Again, it may be tempting to assign variables or something similarly algebraic (e.g. x, x+2, x+4, etc). But instead, how about we take a NARRATIVE and/or GRAPHIC approach? Paint a visual, not unlike the slot method we were using for GMAT combinatorics problems:

___ + ___ +  ___ + ___  =  64

Because these four integers are consecutive odd numbers, we know they are equally spaced. They also add up to a definite sum.

This is where the NARRATIVE approach pays off: if we think about it, there’s only one set of numbers that could fit that description. We don’t even need to calculate them to know this is the case.

You can use a scenario-driven approach with simple numbers to see this. Suppose we use the first four positive odd integers and find the sum:

_1_ + _3_ +  _5_ + _7_  =  16

This will be the only set of four consecutive odd integers that adds up to 16. 

Likewise, let’s consider the next example:

_3_ + _5_ +  _7_ + _9_  =  24

This will be the only set of four consecutive odd integers that adds up to 24. 

It’s straightforward from here to see that for any set of four consecutive odd integers, there will be a unique sum. (In truth, this principle holds for any set of equally spaced integers of any number.) This essentially tells us [for Statement (1)] that once we know that the sum is set at 64 and that the integers are equally spaced, we can figure out exactly what each integer is. Statement (1) is sufficient.

(And notice that I’m not even going to bother finding the integers. All I care about is that I can find them.)

Similarly, let’s take a graphic/narrative approach with Statement (2) by lining the integers up in ascending order:

_ + __ +  ___ + ____  =  64

But very important to note that we must not take Statement (1) into account when considering Statement (2) by itself initially, so we can’t say that the integers are consecutive. 

Here, we clearly represent the smallest integer by the smallest slot, and so forth. We’re also told the largest integer is six greater than the smallest. Now, again, try to resist the urge to go algebraic and instead think narratively. Create a number line with the smallest (S) and largest (L) integers six apart:

S—————|—————|—————|—————|—————|—————L

Narratively, where does that leave us? Well, we know that the other two numbers must be between these two numbers. We also know that each of the four numbers is odd. Every other integer is odd, so there are only two other integers on this line that are odd, and those must be our missing two integers (marked with X’s here):

S—————|—————X—————|—————X—————|—————L

Notice anything interesting? Visually, it’s straightforward to see now that we definitely have consecutive odd integers. Statement (2) actually gives us the same information as Statement (1). Therefore, Statement (2) is also sufficient. The correct answer is D

And again, notice how little actual math we did. Instead, we focused on graphic and narrative approaches to help us focus more on sufficiency, rather than actually solving anything, which isn’t necessary.

Next time, we’ll make a shift to my personal favorite GMAT Number Theory topic: Prime Numbers…

Find other GMAT Number Theory topics here:
Odds and Ends (…or Evens)
Consecutive Integers (plus more on Odds and Evens)
Consecutive Integers and Data Sufficiency (Avoiding Algebra)
GMAT Prime Factorization (Anatomy of a Problem)
A Primer on Primes

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Posted on
24
Mar 2021

Standard Deviation Problem On The GMAT (Normative Distribution)

Standard Deviation 700+ Tips

Hey guys! Today, we’re going to take a look at a standard deviation problem. And standard deviation is a concept that only comes up infrequently on the GMAT. So it’s more important to have a basic understanding of the concepts associated with it than to go really deep. This is true largely for much of the statistics, probability, and combinatorics problems. They show up infrequently until you get to the higher levels and even when you’re at the higher levels, relative to the algebra, arithmetic and even the geometry problems, they play such a small role.

And yet there’s so much math there that it’s very easy to get caught up in spending a lot of time prepping on problems or on these types of math that offer very little in return relative to spending your prep time really mastering the things that come up frequently. I’m not saying don’t learn this stuff I’m saying balance it according to its proportionality on the GMAT. As a general rule you can assume that stat, combinatorics and probability, all that advanced math, constitutes maybe 10 to 15% of what you’ll see on the GMAT. So keep that in mind as you prep.

Problem Language

In this problem the first step is to figure out what the heck we’re actually being asked for and it’s not entirely clear. This one’s written a little bit in math speak. So we have a normal distribution which doesn’t really matter for this problem but if you studied statistics it just means the typical distribution with a mean m in the middle and a standard deviation of d which they tell us is a single standard deviation. So they’re really just telling us one standard deviation but they’re saying it in a very tricky way and they’re using a letter d. If it helps you can represent this graphically.

Graphical Representation

Notice that they tell you something that you may already know: that one standard deviation to either side of the mean is 68 in a normal distribution. This breaks up to 34, 34. But they’re asking for everything below. The +1 side of the distribution. Since the m, the mean is the halfway point, we need to count the entire lower half and the 34 points that are in between the mean and the +d, the +1 standard deviation. This brings us to 84 which is answer choice D. This is primarily a skills problem, that is, you just need to know how this stuff works. There’s no hidden solution path and the differentiation done by the GMAT here is based upon your familiarity with the concept. Rather than heavy-duty creative lifting as we see on so many other problems that have more familiar math, that everyone kind of knows.

I hope this was helpful. Check out below for other stat and cool problem links and we’ll see you guys next time. If you enjoyed this GMAT Standard Deviation problem, try this Data Sufficiency problem next.

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