# Breaking Down Divisibility and Prime Factor Questions

*Help! I understand all the working parts of divisibility and prime factor questions—how to factor, how to work with primes, etc.—but on my last practice exam, it took me so long to figure out what each question was asking that I didn’t finish the section! How do I solve these questions faster?*

*Sincerely,*

*Dumbfounded by Divisibility*

Dear Dumbfounded by Divisibility,

Before you get too down on yourself, plenty of students before you have struggled to understand questions involving factoring and prime numbers. I can absolutely empathize with your frustration, but the good news is that the solution to your efficiency dilemma is relatively straightforward: these problems can be broken down into easy-to-digest pieces. Now, let’s consider the example below:

If and are integers, and , then must be divisible by which of the following?

A) 2

B) 4

C) 5

D) 7

E) 14

The sheer number of possible values for *p* and *r* can be overwhelming, and based on your question, I wouldn’t be surprised if that’s one of the major reasons you have trouble processing the wording of these questions. However, before we get too far ahead of ourselves, let’s establish what information we’re given:

- and are integers

Based on this information, we know that is a perfect square, which means we also know that by association, must be a perfect square as well. Factoring yields . From here, we can see that there are two 2’s but only one 7, and since each perfect square’s prime factors have a partner, we can quickly conclude that any factor for needs to have a partner for 7,making D the correct choice.

As you continue to prepare for the GMAT, tackle tricky questions involving prime numbers by outlining what is given to you. From where I’m sitting, you should rely more on your knowledge of prime numbers and factoring and worry less about learning new “tricks” to help you solve them more quickly. If you’d like to call the few seconds you spend writing down the information you’ve been given a “trick,” then by all means, feel free. However, if your understanding of the properties of prime numbers is as strong as you believe it is, no additional tricks are necessary here.

If you do need a refresher on some of the basic rules of prime numbers, here’s another article we previously posted that outlines the properties you’ll need to know on test day.

Best of luck!

*This post appeared first on the **Economist GMAT Tutor** blog.*