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GMAT Algebra: How to Find Greatest Common Factors

Finding the Greatest Common Factor (GCF) of two numbers is a necessary and useful skill that will enable you to solve a number of different types of GMAT problem solving and GMAT data sufficiency questions. First let’s take a look at the technical aspect of how you actually find the GCF of two numbers, and then we’ll see how this concept is applied on the GMAT.

Finding the Greatest Common Factor (GCF)

The foundational principle you must know to find the Greatest Common Factor of two numbers is Prime Factorization. If you’re not familiar with the technique and usage of Prime Factorization on the GMAT, start by reviewing our article titled “Effectively Using Prime Factorization on the GMAT.”

Now for some definitions.

Greatest Common Factor (GCF): The GCF of two numbers is the largest number that is a factor of both numbers.

Rule (How to find the GCF): The GCF of two numbers is the product of all the numbers that appear in the prime factorizations of both numbers.

What does this mean in plain English? Let’s take a look at this example:

Q: What is the Greatest Common Factor of 24 and 150?

Step 1: Find the prime factorizations of 24 and 150.

GMAT Algebra - Greatest Common Factors

 

In the prime factorization diagram above you see that:
24 = 2 * 2 * 2 * 3
– 150 = 2 * 3 * 5 * 5

Step 2: Identify the overlap of the two numbers’ prime factorizations.

Only the numbers 2 and 3 are factors of BOTH 24 and 150; note that 5 is not a factor of 24.

Step 3: To find the GCF of 24 and 150, apply the rule noted above.

The GCF is “the product of all the numbers that appear in the prime factorizations of both numbers,” in this case one 2 and one 3. Thus, the GCF of 24 and 150 = 2 * 3 = 6.

That’s it! Pretty easy once you get the hang of it.

Application: GCFs on the GMAT

Now that you know how to find the GCF of two numbers, how is this useful on the GMAT? How do we apply what we just learned? Let’s take a look at two application questions, one from GMAT problem solving and one from GMAT data sufficiency.

Question #1 (GMAT Problem Solving)

The greatest common factor of 16 and the positive integer n is 4, and the greatest common factor of n and 45 is 3. Which of the following could be the greatest common factor of n and 210?

(A) 3
(B) 14
(C) 30
(D) 42
(E) 70

The key to all of these types of GMAT algebra questions is to start by finding the prime factorization of the numbers involved. Let’s start with the first premise, which is that the GCF of 16 and n is 4.

The prime factorization of 16 = 2 * 2 * 2 * 2. If the GCF of 16 and n is 4, what does this tell us about n? Well, remembering the rule for finding the GCF of two numbers, that means that n could only have two 2’s in common with 16 as factors, since 2 * 2 = 4.

Looking at the second premise, the prime factorization of 45 = 5 * 3 * 3. What does this tell us about n if the GCF of n and 45 is 3? Certainly that means that n can not have 5 as a factor, and only one 3 as a factor (if it had two 3’s in common, the GCF of n and 45 would be 9, not 3).

What we know about n so far, then, is that it has two 2’s as a factor and 3 as a factor. What else could it have as a factor? Let’s look at the last part of the question, specifically the factors of 210.

The prime factorization of 210 = 2 * 3 * 5 * 7. Right away we see that n has one 2 and one 3 as common factors with 210 so far. N does not have 5 as a factor. There’s nothing in the first two premises of the question to eliminate 7 as a possible factor for n, thus n could have 7 as a factor as well.

Therefore, the GCF of n and 210 could be 2 * 3 * 7 = 42. The answer is D.

Question #2 (GMAT Data Sufficiency)

Is z divisible by 6?
(1) The greatest common factor between 12 and z is 3.
(2) The greatest common factor between 15 and z is 15.

Just because this question is a data sufficiency question instead of a problem solving question, what we’re trying to do when evaluating each statement is essentially the same as we did in answering the problem solving question above.

The major difference of course is that instead of actually having to solve something, we simply have to determine the sufficiency of each statement. Remember from the video lesson on “Data Sufficiency – Part 2,” the first thing you want to do before evaluating the two statements is to ask yourself: “What information, if given to me in the statements, would enable me to answer this question?”

For z to be divisible by 6, the prime factorization of z must at least have a 2 and a 3 in it, right? Thus, when evaluating the statements, all we really need to know is if the statement gives us enough information to determine whether z has both a 2 and a 3 as factors. Let’s take a closer look:

Statement 1: The prime factorization of 12 = 2 * 2 * 3. If the GCF of 12 and z is 3, that means that z does have 3 as a factor, but not 2 (otherwise the GCF would be at least 6). Therefore, the statement is sufficient to definitively answer the question.*

Statement 2: The prime factorization of 15 = 3 * 5. If the GCF of 15 and z is 15, we know that z has a 3 and a 5 as factors, but we don’t know from this statement if it also has 2 as a factor. Therefore we can’t definitively answer the question, so this statement is insufficient.

The answer is A.

* Note: Remember that in Yes/No GMAT Data Sufficiency questions, even if the answer to the question itself is no, the very fact that you were able to make that determination means that you’ve been given enough information in the statement, i.e. the statement is sufficient. Click here to watch a short video explaining Yes/No Data Sufficiency questions in more detail.