Finding factors of integers should become second nature on the GRE; many questions will require you to find the factors of an integer, even if they do not ask you explicitly.

A factor is a divisor, a number that an integer can be evenly divided by. The factors of 8, for example, are 1, 2, 4, 8, -1, -2, -4, and -8. The multiples of an integer x are the infinite products of x and another integer. The multiples of 8 include …-32, -24, -16, -8, 0, 8, 16, 32, 64… and so on.

Prime Factorization: Prime factorization, a.k.a. the factor tree, is a process by which we present an integer as a product of all its primes. The easiest way to do is to make a factor tree. A factor tree is a diagram that breaks down a number into its corresponding factors. Let’s see an example:

Above are the factor trees for 108 and 92. Notice that all the ends of the tree (those numbers that cannot be divided) are primes. So, 108 can be written as 2 x 2 x 3 x 3 x 3, or, more simply, 2² x 3³.  92 can be written as 2 x 2 x 23, or 2² x 23. This practice may seem purposeless, but it has many practical applications.

Factor trees help us simply radicals (answer choices with radicals are almost always in simplified form).

For example, if your answer to a multiple choice question was √96, chances are you won’t see √96 in your answer choices; you’ll probably see the simplified version. To simply a radical, first diagram the factor tree:

So, I know that 96 is the same thing as 2 x 2 x 2 x 2 x 2 x 3.

Since I am trying to simplify the square root, I need to figure out the biggest square in those primes. Because I have five 2s in my primes, I know that the biggest perfect square is 2 x 2 x 2 x 2, or 16, which is 4². Thus, I know that sqrt96 = √16 x√3 x √2. Simplifying this, I know that √96 = 4√6.

Greatest Common Factor: The greatest common factor (GCF) of two or more integers is the greatest integer that is a factor of those integers. For example, the GCF of 24 and 16 is 8, since 8 is the greatest number that is a factor of both. Similarly, the GCF of 60 and 15 is 15. Taking the GCF of bigger numbers, however, is not always so easy. Sometimes, you will be able to arrive at the answer mentally.  When the calculations are more difficult, however, you can use the factor tree to directly arrive at your answer.

Suppose I want to find the GCF of 256 and 72. I’m not quite sure what it is off the top of my head, so I’ll use factor trees to directly arrive at the answer.

Once you perform the prime factorization, it helps to write each as a product of powers: 256 = 2^8 and 72= 2³ x 3². To find the GCF, first find how many primes are common to each prime factorization; in this case, only 2 is common to both. The GCF is the product of all the primes that appear in each factorization, using each prime the smallest number of times in any of the factorizations. In this case, 2³ is smaller than 2^8, so 2³, or 8, is the GCF.

Let’s see another example:

GCF of 68 and 102 and 204.

Using factor trees would yield: 68 = 2² x 17       102 = 2 x 3 x 17      204= 2² x 3 x 17

Here, we have the common factors 17 and 2. 102 has the lowest power of 2, so our GCF is just 2 x 17 = 34. Note: this one would be pretty difficult to figure out without the factor tree.

The best way to get faster at prime factorizations and GCFs is to practice. The good news is, you can practice without looking for sample problems–just make some up on your own.