# Prime Factoring in GMAT Quant: An Overview

There are a lot of **number-properties** problems on the GMAT. Specifically, problems centering on odds/evens and factors/multiples tend to show up frequently on the test.

The good news for test-takers is that these questions rely on a handful of fundamental skills. One skill that you can employ on a variety of problems is **prime factoring.**

## Important definitions

**Prime number **= any number with only 1 and itself as factors (note: 1 itself is not prime)**Composite number** = all non-prime numbers larger than 1

## Applying the concept

**Composite numbers on the GMAT can be separated into prime factors by repeatedly dividing them out until you are left with only primes.**

For example:

12 ÷ 2 equals 6, and 6 ÷ 2 equals 3

So, this means that 12 is 2 • 2 • 3.

From this, we know that 12 has 2, 3, 4, and 6 among its factors (along with 1 and 12).

If we play around with this a bit, we can test for **perfect squares**.**When prime factors come in pairs (e.g. 2 • 2, 3 • 3), the composite number is a perfect square. **

So 12 (2 • 2 • 3) is not a perfect square, but 36 (2 • 2 • 3 • 3) is, since the prime factors of 2 and 3 each come in pairs.

Let's take this concept to a more challenging level:

### Is 1,764 a perfect square?

Try factoring it out.

1,764 factors into 2 • 2 • 3 • 3 • 7 • 7

Now ask yourself: Does each prime factor (2, 3, 7) have a partner?

The answer is yes, so 1,764 is a perfect square.

What is it a perfect square of? Just multiply each partner together:

2 • 3 • 7 = 42

We can also quickly determine **factors** and **multiples** to solve some tough problems.

Example:

### Is 1,386 divisible by 231?

Don’t break out your calculator. Just factor.

231 = 3 • 7 • 11

1386 = 2 • 3 • 3 • 7 • 11

Since 1386 contains all of the prime factors of 231 (plus some extras), it is a multiple of 231.

Keep these prime factoring strategies in mind as you come across lengthy numbers on the GMAT.

**Now, let's apply what you've learned to one of GMAT Tutor's practice questions:**

If p and n are integers, and p^{2}=28n, thennmustbe divisible by which of the following?

(A) 2

(B) 4

(C) 5

(D) 7

(E) 14

Step 1: What is the question asking?

First, we want to know for what, precisely, the question asks. This is a “**must be**” question. We can see that there are lots of possible values for *p* and *r,* but we are asked to find the factor (from among the answer choices) that must be present regardless of our choice of *p* or *r*.

### Step 2: What information helps us solve this problem?

1) p and r are integers

2) p^{2}=28r

### Step 3: What does this information tell us?

From (1), we know that p^{2} is a **perfect square**, and by extension, we know that 28r is a perfect square.

What do we know about perfect squares and their **prime factors**? Each must have a partner!

So, if we factor 28r, we get 2 • 2 • 7 • r. So, whatever other factors *r* contains, it **must**, at least, have the partner for 7. So, r **must** be divisible by 7.

Thus, the answer is D.

Practice making these types of deductions through prime factoring so that on test day you can speed through them.

_{This was a sample of the in-depth instruction that GMAT Tutor offers about solving number-properties questions in the GMAT Quant section. For complete and interactive lessons and online tutor support, subscribe to one of GMAT Tutor's top-rated GMAT prep plans. Commitment-free trials are available for seven days.}