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Welcome back to our series on number properties. This article will cover everything you need to know about the term “greatest common factor” or “greatest common divisor.” These terms refer to the greatest/highest/largest positive integer that is a factor or divisor (by now these terms should appear completely synonymous) of each member of a given set of positive integers.

The greatest common factor (GCF) of a set of integers is the product of the integers’ shared prime factors. To find the GCF of a set of integers, prime factorize each integer, assemble the prime factors that are shared by (or common to) each integer, and multiply these shared primes together. The resulting product is the greatest common factor of the integers in the set.

Before we put the greatest common factor concept to work on an official problem, here’s a practice drill. Try it yourself before looking at the solution that follows.

Find the greatest common factor of 176, 418, and 777.

We need to find the prime factors of each integer:

176 = 24 * 11

418 = 2 * 11 * 19

770 = 2 * 5 * 7 * 11

Hopefully, these stretched your “factor finding” skills a bit. Now, what prime factors are common to each integer? They each have an 11 and at least one 2. 176 has four prime factors of 2, but we can only use one of these 2s to calculate our GCF because the other integers, 418 and 770, each have only one prime factor of 2. Only one prime factor of 2 is “common to” or shared by each integer.

GCF = 2 * 11 = 22

Now let’s look at an official GMAT quant problem involving the greatest common factor:

If n is an integer, what is the greatest common divisor of 12 and n?

1. The product of 12 and n is 432.
2. The greatest common divisor of 24 and n is 12.

(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.

Statement 1 should be easy to recognize as sufficient. If 12n = 432, then we can solve for n. And if we can find n, we can find the GCF/GCD of 12 and n. (n = 36, but you shouldn’t have to find this value to recognize that statement 1 is sufficient.)

Statement 2 is only slightly less simple. The GCD of 24 and n is 12. Remember that this question is asking for the GCD of 12 and n. So since 12 is a factor of n (the data supplied by statement 2), 12 is the GCD of 12 and n. Statement 2 is sufficient. This problem employs a fact we haven’t spoken about much in these articles: every integer has itself as a factor/divisor. So it is possible for the GCD of 12 and some higher integer (like n) to be 12. The correct answer is D.

Here’s one more official GCF/GCD problem:

What is the greatest common divisor of integers r and s?

1. The greatest common divisor of 2r and 2s is 10.
2. r and s are both odd.

(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.

This problem looks simple because it is simple. We can “translate” statement 1 like this: “If r and s each had one more prime factor of 2, their GCD would be 10.” The logical step is to remove a prime factor of 2 from 10 via division: 10 / 2 = 5. So 5 is the GCD of r and s. Statement 1 is sufficient.

Statement 2 is clearly insufficient by itself, but it might tempt you to second-guess your conclusion about statement 1. Don’t be fooled. We can already deduce from statement 1 that r and s are both odd, and this data isn’t necessarily helpful anyway. The correct answer is A.

To recap: the greatest common factor/divisor of a set of integers is the product of the integers’ shared primes. In the next article, we’ll address the related topic of the least common multiples.

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Contributor: Elijah Mize (Apex GMAT Instructor)