One of the most fundamental and widely applicable skills for GMAT quant is the ability to break down integers to their prime factors. Integers are a class of numbers sometimes referred to as “whole numbers.” Integers can be positive, negative, or 0 in value, and they can be written without the use of fractions or decimals.
Prime number divisibility
An integer’s prime factors are the prime numbers that, when multiplied together, produce the integer. A prime number is an integer that is not divisible by any integers other than itself and 1. Divisibility means that when an integer is divided by another integer, the result (called the quotient) is also an integer. Here are some examples:
56 / 7 = 8 → 56 is divisible by 7
56 / 6 = 9.33 → 56 is not divisible by 6
132 / 12 = 11 → 132 is divisible by 11
132 / 15 = 8.8 → 132 is not divisible by 15
65 / 13 = 5 → 65 is divisible by 13
65 / 8 = 8.125 → 65 is not divisible by 8
If integer n is divisible by integer x, then integer x is called a divisor of integer n. Integer x may also be called a factor of integer n. The term “divisor” is, of course, related to the operation of division, and the term “factor” is related to the operation of multiplication. Relationships between integers may be spoken of in terms of either division or multiplication. These represent two interchangeable ways of speaking about integers.
If we want to speak in terms of multiplication, then instead of saying that integer n is divisible by integer x, we say that integer n is a multiple of integer x. This means that there is some integer by which x can be multiplied to produce n. Here are the same examples from before, this time in terms of multiples.
56 / 7 = 8 → 56 is a multiple of 7 (and a multiple of 8)
56 / 6 = 9.33 → 56 is not a multiple of 6
132 / 12 = 11 → 132 is a multiple of 12 (and a multiple of 11)
132 / 15 = 8.8 → 132 is not a multiple of 15
65 / 13 = 5 → 65 is a multiple of 13 (and a multiple of 5)
65 / 8 = 8.125 → 65 is not a multiple of 8
Note the parentheticals: we could rearrange the values to say 56 / 8 = 7 or 132 / 11 = 12 or 65 / 5 = 13. To generalize this principle, if integer n is a multiple of integer x, then integer n is also a multiple of whatever integer, when multiplied by x, produces n.
Here are five different ways to ask the same question about two integers, n and x:
Is n divisible by x?
Is n a multiple of x?
Is x a divisor of n?
Is x a factor of n?
Is n/x an integer?
As you prepare to take the GMAT, you should see so many of these questions that you understand what is being asked without really noticing the changes in terminology. The goal is for your brain to develop a concept of divisibility that is readily identified by any of the above ways of speaking.
The frequency of this concept on GMAT quant makes it hard to get a great score without knowing your multiplication table like the back of your hand. If there are any two integers between 1 and 12, inclusive, whose product (the result when these integers multiply each other) you don’t know instantly, it’s time to review your multiplication table and possibly whip out the flash cards. As elementary as this feels, it will serve you well.
The first 15 prime numbers
Now to return to prime factors. (We could call these prime divisors if we liked, but to do so would be rather unconventional.) Again, a prime number is an integer that is not divisible by any integers other than itself and 1. It’s worth pointing out that every integer is divisible by itself and 1. Every number, whether it’s an integer or not, is divisible by itself, since x/x = 1. (The case of 0/0 is best left to professional mathematicians; it won’t appear on the GMAT.) And dividing any number, whether it’s an integer or not, by 1, effectively does nothing, just as multiplying any number by 1 effectively does nothing. x/1 = x, and x*1 = x.
Here are the first 15 prime numbers, the ones occurring between 1 and 50:
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47
Try dividing any of these numbers by an integer other than itself or 1, and your quotient won’t be an integer. (If you’re curious why these and not any other integers represent the first 15 primes, you’ll have a hard time finding an explanation, because a mathematically provable model for describing or predicting the series of prime numbers remains one of the most elusive mysteries in mathematics.) Note that every prime number except for 2 is odd.
With this understanding of prime numbers in place, it follows that the prime factors of an integer are the integer’s “building blocks” or fundamental elements, which cannot themselves be further broken down. Let’s use some official GMAT problems to practice the process of reducing integers to their prime factors. We refer both to this process and to the resulting list of primes as the “prime factorization” of the integer.
The “prime sum” of an integer n greater than 1 is the sum of all the prime factors of n, including repetitions. For example, the prime sum of 12 is 7, since 12 = 2 * 2 * 3 and 2 + 2 + 3 = 7. For which of the following integers is the prime sum greater than 35?
This problem is helpful because it includes an example of prime factorization with the integer 12. There are two possible “routes” for us to perform the prime factorization of 12. We can work through and display these “routes” with a factor tree tool.
When we perform a prime factorization of an integer, we “break down” the number into sets of factors until all of the factors are prime and cannot be further broken down. With the integer 12, we may begin with the factor pair of 2 * 6 or with the factor pair of 3 * 4. In the first case, we finish by breaking the 6 into 2 * 3. In the second case, we finish by breaking the 4 into 2 * 2. Either way, the resulting prime factorization of 12 is 2 * 2 * 3. (I like to underline the prime factors as I find them in order to keep track.)
No matter which “route” is taken in the prime factorization of an integer, the resulting list of prime factors is always the same. It is often helpful to group repetitions of the same prime factor using exponents. So for 12, instead of “2 * 2 * 3,” we can express the prime factorization as “22 * 3.”
In this problem, we are looking for the number whose “prime sum” – or, as we are told, the sum of all its prime factors – is greater than 35. If you know your first 15 prime numbers, a certain answer choice here jumps out. But for the sake of learning and practice, we will prime factorize each answer choice and check its prime sum. Here’s answer choice A:
Prime factorization: 23 * 5 * 11
Prime sum = 2 + 2 + 2 + 5 + 11 = 24
When an integer ends with one or more 0s (as 440 does), it’s best to begin by isolating the factors of 10, each of which breaks into the prime factors of 2 and 5. Here’s how this would work for a larger integer:
The prime sum for 440 was 24, which is too low, so we’ll have to move on to answer choice B:
This is a prime example (pun intended) of why you should memorize some powers of 2, 3, 5, and 6 along with your multiplication table! When you see a number like 512, it can be a big time-saver to know immediately, “Oh, that’s 29.” The prime factorization of 512 is 29, and its prime sum is 18. This is too low, so we’ll move on to answer choice C:
Prime factorization: 22 * 5 * 31
Prime sum: 2 + 2 + 5 + 31 = 40
C is the correct answer, because 620 contains the prime factor 31, virtually guaranteeing that the prime sum is greater than 35. (Given the importance of primes on GMAT quant, perhaps it is no coincidence that the section contains 31 questions.)
Official GMAT problem for practice
Here are the prime factorizations of answer choices D and E, just for practice:
Prime factorization is one of the easiest GMAT quant skills to practice because you can use any integer that comes to mind! Let’s try another official GMAT problem:
How many prime numbers between 1 and 100 are factors of 7,150?
This is a very straightforward problem. Just prime factorize 7,150 and see how many different prime factors there are.
Straightforward, but easier said than done. If you got stuck trying to divide 715, you can always use an “obvious factor” like 5. Since 715 has a units digit of 5, we know that it is divisible by 5. Then all you need is the quotient of 715 / 5.
Dividing 143 is a little trickier, since it has no obvious factors. If you mistakenly conclude that 143 is itself a prime factor, you will choose incorrect answer choice B for this problem. Sharp factoring eyes will notice that 143 = 130 + 13, or 11 * 13. We can count four different prime factors of 7,150 (2, 5, 11, and 13), so the correct answer choice is D. If this was challenging for you, remember, you can practice with any integer.
Here’s a final problem for this article:
If 3 < x < 100, for how many values of x is x/3 the square of a prime number?
The initial step to solving this problem is less obvious. We want to know how many integers greater than 3 and less than 100 are divisible by 3 and then by the square of a prime number, with no further prime factors. The only values that can work are the multiples of 3 (since these are the only values divisible by 3), but there are too many multiples of 3 between 3 and 100 to check! The best way to proceed is to reverse the process. Square a prime, multiply your result by 3, and see if this final product is less than 100.
22 * 3 = 12
Hey, we know this one from before. Let’s keep going:
32 * 3 = 9 * 3 = 27
52 * 3 = 25 * 3 = 75
72 * 3 = 49 * 3 = 147
This is already too high, so only three values (12, 27, and 75) satisfy the conditions specified in the problem. The correct answer choice is B.
This concludes our introduction to number properties and prime factors. In the next article, we’ll learn about the factors of perfect squares.
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Contributor: Elijah Mize (Apex GMAT Instructor)