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The Key to GMAT “Number Theory” Questions

Multiples factors and divisibility on the GMATYou’re likely to encounter at least a couple of what I call “number theory” questions on the GMAT, on both the Problem Solving and Data Sufficiency sections of the test. They can be phrased in various ways and ask about factors, multiples, divisors, or sometimes straight-out divisibility itself. The good news for you is that the key to answering these types of questions is virtually the same in all cases. But first, a few definitions so that we’re all on the same page.

Number Terminology and Definitions

  1. Factors: A factor is an integer that can be divided evenly into another integer (“divided evenly” means that there is no remainder). Example: Factors of 12 are 1, 2, 3, 4, 6, and 12. Any non-zero integer has a finite number of factors.
  2. Multiples: A multiple is a number that results from a given integer being multiplied by another integer. Example: Multiples of 12 include 12, 24, 36, 48, etc. Proof: 12 x 1 = 12, 12 x 2 = 24, 12 x 3 = 36, and 12 x 4 = 48, etc. Any non-zero integer has an infinite number of multiples.
  3. Divisor: A number by which another number, the dividend, is divided. In the expression 9 ÷ 3, 3 is the divisor.
  4. Dividend: A number that is to be divided by another number, the divisor. In the expression 9 ÷ 3, 9 is the dividend.
  5. Prime number: Prime numbers are non-negative integers which have two and only two factors; that is, the factors 1 and themselves. The first 10 primes are 2, 3, 5, 7, 11, 13, 17, 19, 23 and 29. Note that 1 is not a prime number as it only has one factor, i.e., 1. Also, the number 2 is not only the smallest prime but also the only even prime number.
  6. Prime factorization: Prime factorization is simply the process of breaking a number down into its most basic, prime number factors. For an article that explains prime factorization in more detail including how it’s used on the GMAT, click here.
  7. Least Common Multiple (LCM): The least common multiple of two or more numbers is the smallest number that is divisible by those original numbers. For an article that explains LCM’s in more detail including how it’s tested on the GMAT, click here.
  8. Greatest Common Factor (GCF): The greatest common factor of two numbers is the largest number that is a factor of both numbers. For an article that explains GCF’s in more detail including how it’s tested on the GMAT, click here.

The Key to Solving “Number Theory” Questions

The key to solving virtually all questions about divisibility on the GMAT is factoring, and specifically, working down to prime factors (i.e. prime factorization). The reason this is important is that prime factorization gives you the “building blocks” that comprise the numbers in question. It gives you the ability to compare apples to apples, in other words.

How to use prime factorization to solve GMAT questions

Figure 1: Prime Factorization of 36

Take the number 36, for example. Prime factorization of 36 (Figure 1) reveals that the “building blocks” of 36 are two 2’s and two 3’s.

How is this useful? Specifically, how can we know if a certain number is a multiple of 36? If it contains all of the same “building blocks” as 36, then it will by definition be a multiple of 36.

Rule #1: If one variable (let’s call it x) contains all of the same factors as another variable (call this one y), then by definition, x is a multiple of y. This always works because y itself is always a factor of y, so if x contains y as a factor, then x is a multiple of y.

For example, is 72 a multiple of 36? Well, the “building blocks” (i.e. prime factorization) of 72 are 2 x 2 x 2 x 3 x 3. So even though 72 contains an extra “2” building block, it at least contains all of the same building blocks as 36, and therefore it is a multiple of 36.

But what about a number like 450? Is 450 a multiple of 36? As you would expect, let’s start by factoring 450. The “building blocks” of 450 are 2 x 3 x 3 x 5 x 5. In this case, 450 shares some of the same factors as 36. In fact, it has one 2 and two 3’s as common factors with 36. But it doesn’t contain all of the same factors as 36, and therefore it’s not a multiple of 36.

Rule #2: If one variable (let’s call it a) contains the same basic prime factors as another variable (call this one b), then we don’t know whether a is a multiple of bIt’s possible, such as when a = 12 and b = 6; b has 2 and 3 as prime factors, and a shares that characteristic. In this case, 12 is a multiple of 6. On the other hand, if a = 30 and b = 12, then b has 2 and 3 as prime factors, and a shares that characteristic, but now, 30 is not a multiple of 12. This is because b actually contains two 2s, but 30 contains only one 2.

By the way, a multiple of a certain number can contain additional factors that aren’t included as factors of that number. For example, the number 900 is a factor of 36 because its building blocks are two 2’s, two 3’s, and two 5’s. In this case 900 has a factor (5) that 36 doesn’t have. But as long as it contains all of the factors that 36 contains, that’s okay, it’s still a multiple of 36.

Basic Divisibility Rules

Occasionally it’s helpful to know if a number is flat-out divisible by another number. It may be worth memorizing these “tricks” to quickly discern whether or not a number is divisible by the numbers 1 through 10, as seen in this chart:

Divisibility rules - determine whether one number is divisible by another number on the GMAT

Data Sufficiency Application Example

With this core understanding about factors, multiples, prime factorization, and divisibility in mind, try your hand at this sample GMAT Data Sufficiency question:

m is a multiple of 13. Is mn a multiple of 195?

(1) n has every factor that 45 has
(2) m is divisible by 18

Here’s a detailed explanation of the answer to this question as posted on Beat the GMAT:

“First, let’s make sure we are clear on what we’re looking for. This is a Yes/No Data Sufficiency problem, so it doesn’t matter whether mn is a multiple of 195 or not. That’s not what determines sufficiency or insufficiency. The key is whether we know for certain one way or the other. If we’re certain mn is a multiple of 195, that’s sufficient. If we’re certain mn is not a multiple of 195, that’s also sufficient. The data is insufficient only if we can’t tell.

We know m is a multiple of 13, but there’s not much to do with that, so let’s focus on the other number in the problem. The question wants to know whether mn is a multiple of 195, which is another way of asking whether 195 is a factor of mn, which is another way of asking whether mn is divisible by 195. So how do we figure out whether a number is divisible by 195 or not? By breaking it down into prime factors. 195 ends with a 5, so it must be divisible by 5, and we can start there. 195 ÷  5 = 39. We can break down 39 a little more. 39 is 3 × 13. So the prime factorization of 195 is 5 × 3 × 13. What this means is that any number that is divisible by 5 and by 3 and by 13 is thereby divisible by 195. So returning to the question, we now have a way of figuring out whethermn is a multiple of 195. If mn is divisible by 5, 3, and 13, then it will be a multiple of 195. If mn is lacking any of those factors, then it won’t be. The question actually becomes even a little simpler, because we already know that m is a multiple of 13 — that’s given to us at the start. So the 13 is taken care of, and when we turn to the two statements we only need to look for the 3 and the 5. Let’s do that.

1) has every factor that 45 has.

Does this tell us for certain whether mn is divisible by 3 and 5? Since 45 is divisible by 3 and 5, then n must be, and so must mn. Thus, Statement 1 is sufficient and we can eliminate answer choices B, C, and E.

2) is divisible by 18.

Does this tell us for certain whether mn is divisible by 3 and 5? Well 18 is divisible by 3, but not by 5. So although we know that we have the 3 we’re looking for, we don’t know anything about the 5. It’s important to recognize that Statement 2 has not established that mn lacks 5 as a factor. We just don’t know. There’s nothing that prevents from being divisible by 18 and also divisible by 5. Or could be divisible by 5. Because we are ignorant about the 5, Statement 2 is insufficient, and the correct answer is A.”

Problem Solving Application Example

Divisibility can also be tested on GMAT problem solving questions as seen in this example. Give it a try and post your answer (or questions) in the comment box below.

Q: If N is a positive integer, then the least value of N for which N! is divisible by 1,000 is ?

(A) 1
(B) 4
(C) 9
(D) 15
(E) 30

If you get stuck or to see a detailed explanation of exactly how to solve this question (with the underlying rationale), watch this video: