Posted on
07
Sep 2021

## How-to GMAT: No Calculator? Use These Mental Math Tips Instead

The GMAT is an exam largely focused on numbers and numerical data. And while doing math on the GMAT should be avoided sometimes it is inevitable. True, the test-taker is given a calculator for the duration of the Integrated Reasoning section but the same cannot be said for the Quantitative Reasoning Section.

###### The Basics

Before explaining any methods for dividing and multiplying with ease, let’s make sure we have revised a few simple rules:

• Numbers with an even last digit are divisible by 2 – 576 is and 943 is not;
• Numbers with a sum of digits divisible by 3 are also divisible by 3 – 3,465 for example (3+4+6+5=18);
• If the last 2 digits of a number a divisible by 4, the number itself is divisible by 4 – 5,624 for example (because 24/4=6);
• Numbers with last digit 0 or 5 are divisible by 5;
• Numbers that can be divided by both 2 and 3 can be divided by 6;
• Similar to numbers divisible by 3, numbers divisible by 9 must have a sum of digits divisible by 9 – 6,453 for example;
• If the last digit of a number is 0 it is divisible by 10;

With that out of the way, we can move onto some more advanced mental math techniques.

###### Avoid division at all costs

Don’t divide unless there is no other option. And that is especially true with long division. The reason why long division is so perilous is that it is very easy to make a careless mistake as there are usually several steps included in the calculation, it takes too much time, and to be honest, few people are comfortable doing it.

Fortunately, the GMAT doesn’t test the candidates’ human-calculator skills but rather their capacity to think outside the box and show creativity in their solution paths, especially when under pressure – exactly what business schools look for.

However, sometimes you cannot avoid division, and when that is the case remember: Factoring is your best friend. Always simplify fractions especially if you’ll need to turn them into decimals. For example, if you have 234/26 don’t start immediately trying to calculate the result. Instead, factor them little by little until you receive something like 18/2 which is a lot easier to calculate.

A tip for factoring is to always start with smaller numbers as they are easier to use (2 is easier to use compared to 4, 6, or 8) and also look for nearby round numbers.

If you have to calculate 256/4 it would be far less tedious and time-consuming to represent 256 as 240+16 and calculate 240/4+16/4=60+4=64. Another example is 441/3. If we express it like 450-9 it is far easier to calculate 450/3-9/3=150-3=147.

###### Dividing and Multiplying by 5

Sometimes when you have to divide and multiply by 5 (you’ll have to do it a lot) it would be easier to substitute the number with 10/2. It might not always be suitable for your situation but more often than not it can be utilized in order to save some time.

###### Using 9s

With most problems, you could safely substitute 9 with 10-1. For example, if you have to calculate 46(9) you can express it as 46(10 – 1) which is a lot more straightforward to compute as 46(10) – 46(1) = 460 – 46 = 414

You can also use the same method for other numbers such as 11, 8, 15, 100, etc:

18(11) = 18(10 + 1) = 180 + 18 = 198

28(8) = 28(10 – 2) = 280 – 56 = 224

22(15) = 22(10 + 5) = 220 + 110 = 330

26(99) = 26(100 – 1) = 2600 – 26 = 2574

###### Dividing by 7

The easiest way to check if a number is divisible by 7 is to find the nearest number you know is divisible by 7. For instance, if you want to check if you can divide 98 by 7 you should look for the nearest multiple of 7. In this instance either 70, 77, or 84. Start adding 7 until you reach the number: 70 + 7 = 77 + 7 = 84 + 7 = 91 + 7 = 98. The answer is yes, 98 is divisible by 7 and it equals 14

###### Squaring

When you have to find the square of a double-digit number it might be easier to break the number into components. For example, 22^2 would be calculated like this:

22^2
= (20 + 2)(20 + 2)
= 400 + 40 + 40 + 4
= 484

Similarly, if you have to find the square of 39 instead of calculating (30 + 9)(30 +9) you could express it like this:

39^2
= (40 – 1)(40 – 1)
= 1600 – 40 – 40 + 1
= 1521

You can use the same approach when multiplying almost any double-digit numbers, not only squaring. For example 37 times 73:

(40 – 3)(70 + 3)
= 2800 + 120 – 210 – 9
= 2701

###### Conclusion

This ends the list of mental math tips and tricks you can utilize to make the Quant section a bit less laborious. Keep in mind that no strategy or shortcut would be able to compensate for the lack of proper prep so it all comes down not only to practicing but doing so the right way.

For more information regarding the GMAT Calculator, GMAT Calculator & Mental Math – All You Need To Know, is a very insightful article to read.

Posted on
27
Jul 2021

## GMAT Calculator & Mental Math – All You Need To Know

Author: Apex GMAT
Contributor: Ilia Dobrev
Date: 27 July, 2021

Are you allowed to use a calculator on the GMAT? While this may seem as a pretty straightforward question to answer, it does deserve a separate blog post as it hides details that are vital for achieving a top GMAT score not only on the quantitative section, but on the exam as a whole.

Well, the answer is both Yes and No. This article aims to examine the different scenarios where you are allowed to use such a device and how you can make full use of its potential. But, if you are used to doing math with a calculator, do not worry as we have compiled a list of some mental math techniques that you can use to your advantage and even save much important time while still being spot-on with your answers.

#### Calculator on the GMAT | Explained

• You are not allowed to bring your own calculator to the GMAT exam.
According to the GMAC, no personal items are allowed in the exam room of any of the certified test centers.
• You cannot use a calculator on the Quantitative section of the GMAT.
Despite the fact that we are so used to using calculators to help us with arithmetic operations, you should not feel intimidated that you are not allowed to use any type of calculator on the GMAT Quantitative section. However, you will be provided with a blank canvas by the proctor of the exam where you will have plenty of space to practice to manually compute any calculations, if needed.

You should not worry as the GMAT exam is not designed to test you on complicated mathematical operations or complex calculations. Instead, the quant section draws from secondary-level math skills like basic algebra and geometry, which are mastered in high school, to test other kinds of abilities like critical thinking, logical reasoning, and problem-solving. In fact, the majority of the Quant questions can (and should) be answered without any calculations beyond estimation. A typical example of how you can use mental math to get to the right answer while saving precious time on the GMAT is the Movie Night combinatorics problem. Another type of common GMAT quant questions are data sufficiency problems, which are also more about reasoning than  calculations. You’ll only need to do basic calculations and can rely on estimation for anything more complicated. If you have to do the math, the GMAC usually keeps the numbers simple and avoids decimals. When you see large numbers or complex fractions, then it’s a good bet that there’s an easier solution path to embark on other than calculating.

Surprisingly or not, a calculator will be provided for use during the GMAT Integrated Reasoning section of the test. This GMAT calculator has the standard basic functions, CE (clear entry) button, C (clear) button, an sqrt function, a % (percentage) button, and a 1/x button that calculates the reciprocal of the entry currently on the screen. Also, there is a row with the standard memory functions

• MS (memory store) stores the current entry in the calculator’s memory for subsequent use.
• MR (memory recall) displays the latest number stored in the calculator’s memory so that it can be used for the next calculations.
• M+ (memory addition) adds the current entry to the value that is currently stored in the calculator’s memory. This button is helpful when you need to add a long series of numbers and don’t want to retype each one.
• MC (memory clear) erases whatever is in the current memory. You should click this button before every new calculation scenario.

### Survival Tips & Tricks

#### Do not overuse the IR calculator.

While you are provided with a basic GMAT calculator during the Integrated Reasoning section, you might not want to use it too often as you’ll waste more time than you’ll save. You can also apply the solution paths you are using in the Quant section to some problems in the Integrated Reasoning section.

#### Constantly practice Mental Math operations.

A huge morale boost is that mental math operations are easy to learn with some practice. You can add, multiply, subtract, and divide when you pay bills, check out at the grocery store, calculate a tip, etc. without using a calculator.

#### Make accurate estimations

The key to saving a considerable amount of time on the GMAT exam is efficiency in estimations. Transform numbers to less unwieldy figures like 0 or 5 for the purpose of calculations. You can then browse the answer choices to see which is closest to your preliminary estimate.

#### Do not use a calculator when you are prepping for the GMAT quant section.

This is a great way to practice mental math operations outside the daily life operations. The test setting and quant context will let you get used to this environment so that you know what to expect on test day.

#### Familiarize yourself with a basic GMAT calculator and do use its Memory functions.

As this will be your only technical aid during the GMAT Integrated Reasoning section, you’d better spend some time making the most out of it. Especially when you are pressed by time, memorizing calculated values for further operations in the calculator’s memory can be crucial for staying on track with a healthy exam pace.

#### Guide yourself by looking at the answer choices.

Looking at the answer choices can immediately permit you to eliminate a couple of options. Even if you are pushed by time, you can easily make a more educated decision depending on your reasoning that will boost the chances of picking the correct answer.

#### Do not freak out if you see large numbers.

Remember that the people who stand behind the GMAT are aware that they are designing questions that are supposed to be answered without using a calculator. This also keeps the arithmetic from being too difficult and gives you the opportunity to apply a more straightforward approach.

Posted on
21
Jul 2021

## 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 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.

Posted on
14
Jul 2021

## GMAT Geometry Problem

Hey guys, top level geometry problems are characterized typically by stringing a whole bunch of different rules together and understanding how one thing relates to the next thing, to the next thing. Until you get from the piece of information you started with to the conclusion. We’re going to start out by taking a look at this problem using the z equals 50° and seeing how that information goes down the line.

But afterwards we’re going to see a super simple logical pathway utilizing a graphic scenario that makes the z equals 50° irrelevant. To begin with we’re being asked for the sum of x and y and this will come into play on the logical side. We need the sum not the individual amounts but let’s begin with the y. We have a quadrilateral and it has parallel sides which means the two angles z and y must equal 180°. That’s one of our geometric rules. If z is 50° that means y is 130° and we’re halfway there.

Next we need to figure out how x relates and there are several pathways to this. One way we can do it is drop. By visualizing or dropping a third parallel line down, intersecting x, so on the one hand we’ll have 90 degrees. We’ll have that right angle and on the other we’ll have that piece. Notice that the parallel line we dropped and the parallel line next to z are both being intersected by the diagonal line going through which means that that part of x equals z. So we have 50° plus 90° is 140°. 130° from the y, 140° from the x, gives us 270°.

Another way we can do this is by taking a look at the right triangle that’s already built in z is 50° so y is 1 30°. now the top angle in the triangle must then be 180° minus the 130° that is 50°. it must match the z again we have the parallel lines with the diagonal coming through then the other angle the one opposite x is the 180° degrees that are in the triangle minus the 90° from the right triangle brings us to 90° minus the 50° from the angle we just figured out means that it’s 40° which means angle x is 180° flat line supplementary angles minus the 40° gives us 140° plus the 130° we have from y again we get to 270°.

#### Graphic Solution Path

Now here’s where it gets really fun and really interesting. We can run a graphic scenario here by noticing that as long as we keep all the lines oriented in the same way we can actually shift the angle x up. We can take the line that extends from this big triangle and just shift it right up the line until it matches with the y. What’s going to happen there, is we’re going to see that we have 270° degrees in that combination of x and y and that it leaves a right triangle of 90°, that we can take away from 360° again to reach the 270°.

Here the 50° is irrelevant and watch these two graphic scenarios to understand why no matter how steep or how flat this picture becomes we can always move that x right up and get to the 270°. That is the x and y change in conjunction with one another as z changes. You can’t change one without the other while maintaining all these parallel lines and right angles. Seeing this is challenging to say the least, it requires a very deep understanding of the rules and this is one of those circumstances that really points to weaknesses in understanding most of what we learn in math class in middle school, in high school. Even when we’re prepping only scratches the surface of some of the more subtle things that we’re either allowed to do or the subtle characteristics of rules and how they work with one another and so a true understanding yields this very rapid graphic solution path.

#### Logical Solution Path

The logical solution path where immediately we say x and y has to be 270° no matter what z is and as you progress into the 80th, 90th percentile into the 700 level on the quant side this is what you want to look for during your self prep. You want to notice when there’s a clever solution path that you’ll overlook because of the rules. Understand why it works and then backtrack to understand how that new mechanism that you discovered fits into the framework of the rules that we all know and love. Maybe? I don’t know if we love them! But they’re there, we know them, we’re familiar with them, we want to become intimate. So get intimate with your geometry guys put on some al green light some candles and I’ll see you next time.

If you enjoyed this problem, try other geometry problems here: GMAT Geometry.

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!

Posted on
13
May 2021

## 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.

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.

Posted on
20
Apr 2021

## Intro to GMAT Data Sufficiency- All you’ll need to know

By: Apex GMAT
Contributor: Altea Sulollari
Date: 20th April 2021

As a GMAT test-taker, you are probably familiar with data sufficiency problems. These are one of the two question types that you will come across in the GMAT quant section, and you will find up to 10 of them on the exam. The rest of the 31 questions will be problem-solving questions.

The one thing that all GMAT data-sufficiency questions have in common is their structure. That is what essentially sets them apart from the problem-solving questions.

Keep on reading to find out more about these questions’ particular structures and the topics that they cover:

## The question structure:

The GMAT data sufficiency problems have a very particular structure that they follow and that never changes. You are presented with a question and 2 different statements. You will also be given 5 answer choices that remain the same across all data sufficiency problems on the GMAT exam. These answer questions are the following:

A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
B)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D) EACH statement ALONE is sufficient.
E) Statements (1) and (2) TOGETHER are NOT sufficient.

Your job would be to determine whether the 2 statements that you are provided with are sufficient to answer the question.

## What topics are covered?

Some of the math topics that you will see in this type of question are concepts from high school arithmetic, geometry and algebra.

Below, you’ll find a list of all concepts you need to know for each math topic:

### Geometry

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

### Algebra

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

### Arithmetic

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

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

## Common mistakes people make when dealing with this question type

### Actually solving the question

This is the #1 mistake most test-takers make with these problems. These problems are not meant to be solved. Instead, you will only need to set up the problem and not execute it. That is also more time-efficient for you and will give you some extra minutes that you can use to solve other questions.

### Over-calculating

This relates to the first point we made. This question type requires you to determine whether the data you have is sufficient to solve the problem. In that case, calculating won’t help you determine that. On the contrary, over-calculating will eat up your precious minutes.

### Rushing

This is yet another common mistake that almost everyone is guilty of. You will have to spend just enough time reading through the question in order to come up with a solution. Rushing through it won’t help you do that, and you will probably miss out on essential details that would otherwise make your life easier.

### Not understanding the facts

What most test-takers fail to consider is that the fact lies in the 2 statements that are included in the questions. Those are the only facts that you have to consider as true and use in your question-solving process.

## 3 tips to master this question type:

### Review the fundamentals

That is the first step you need to go through before going in for actual practice tests. Knowing that you will encounter these high school math fundamentals in every single quant problem, is enough to convince anyone to review and revise everything beforehand.

This might sound a bit intimidating at first as most answer choices are very long sentences that tend to be similar to each other in content. However, there is a way to make this easier for you. What you need to do is synthesize the answer choices into simpler and more manageable options. That way, they will be easier to remember. This is what we suggest:

A) Only statement 1
B)
Only statement 2
C) Both statements together
D) Either statement
E) Neither statement

### Examine each statement separately

That is definitely the way to go with this GMAT question. You will need to determine whether one of the statements, both, either, or neither is sufficient, and you cannot do that unless you look at each of them separately first.

Now that you have read the article and are well-aware of the best ways to solve data sufficiency problems on the GMAT, try your hand at this question: Number Theory: Data Sufficiency

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:

Notice 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:

Now, 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:

Notice 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.

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

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

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…