GRE Exam: A Simple Guide To Number Properties - Factors and Multiples

On a broad level, you can think of properties of numbers as the branch of math concerned with how numbers behave in certain situations.

Though this is an enormous field in formal mathematics, the GRE will be concerned with properties of numbers in the following contexts: divisibility, odds and evens, positives and negatives, and evenly spaced sets.

Because these areas are all concerned with concrete mathematical rules and what you can deduce from these rules, questions testing these concepts will often appear in Quantitative Comparison questions or a “must be true” or “could be true” format in Discrete Quantitative questions.

Factors and Multiples
Any whole number is an integer. For example, 2 and –9 are integers, but and –7.2 are not. The factors (or divisors) of an integer are the integer values that divide evenly into that number. 2 is a factor of 12 because , which is an integer.

But 5 is not a factor of 12, because , which is not an integer.

To determine the factors of a number, you can create a factor table.

For example, the factors of 12 are:

The multiples of an integer are the products that result when that integer is multiplied by another integer.

For example, the multiples of 12 are 12(1) = 12, 12(2) = 24, 12(3) = 36, . . . .

Note that multiples and factors are essentially opposites of each other.

Since 6 is a factor of 24, 24 is a multiple of 6.

The GRE expresses the preceding relationships in several ways.

All of the following sentences mean the same thing:

Many test-takers tend to confuse factors and multiples. If this is the case, think to yourself that there are finite factors and many multiples.

The factors are what create a number and the multiples are what result from that number.

Prime Factors and the Factor Tree
Any number will always have 1 and itself as divisors. If an integer is divisible only by 1 and itself, then it is a prime number. Examples of prime numbers are 2, 3, 5, 7, 11, 13, and so on.
1 is not a prime number, and 2 is the only even prime number!

A prime factor is any factor of an integer that is also prime.

For example, 2 and 3 are prime factors of 12, but 4 is not.

There are two important properties about prime factors:
1. Any integer can be expressed as the product of its prime factors.

For example: 12 = 2 × 2 × 3.
2. The factors of any integer will be completely determined by the prime factors of that integer.

For example, 12 = 2 × 2 × 3.

The factors of 12 are 1, 2, 3, (2 × 2), (2 × 3), (2 × 2 × 3).

To determine the prime factors of a number, you should create a factor tree.

The following is the factor tree for 240.

The prime factorization of 240 is thus: (24) × 3 × 5. From Property 2 earlier, you can infer that 40 is a factor of 240 but that 32 is not. Why? Because the prime factors of 40 (2 × 2 × 2 × 5) are contained in the prime factorization of 240, but the prime factors of 32 (2 × 2 × 2 × 2 × 2) are not contained in the prime factorization of 240. One important principle that extends from the preceding explanation is the following:
If a is a factor of b, and b is a factor of c, then a must be a factor of c.
For example, since 40 is a factor of 240, 8 and 5 (which are factors of 40) must also be factors of 240.

Generally, when doing questions that concern divisibility, you should focus on prime factorization.

If y is divisible by 12, which of the following must be true? Indicate all that apply.
y is divisible by 24
y is divisible by 6
y is divisible by 4

SOLUTION: If y is divisible by 12, then the prime factors of 12 must be prime factors of y. Create a factor tree to determine the prime factors of 12.

You can therefore infer that y has 2, 2, and 3 in its prime factorization.

Since y has 2 × 2 in its prime factorization, y must be divisible by 4. Since y has 2 × 3 in its prime factorization, y must be divisible by 6. The correct answer is B and C.

Greatest Common Factor and Least Common Multiple
The greatest common factor (GCF) of a set of numbers is the largest integer that divides evenly into all the numbers. To determine the greatest prime factor of a set of numbers, break each of the numbers down into their prime factors and circle the shared factors. The product of the shared factors will be the GCF.

For this question, write your answer in the box.
What is the greatest common factor of 12, 72, and 88?


SOLUTION: Determine the prime factors of each of the numbers and circle their common prime factors:

12, 72, and 88 each have two 2s in their prime factorizations. The GCF of the three numbers is thus 2 × 2 = 4.
The least common multiple (LCM) of a set of numbers is the smallest integer that is divisible by all the numbers in the set. The LCM must therefore contain the prime factors of each number in the set. As with the GCF, prime factorization is important for LCM questions.

If x is the smallest integer that is divisible by 9, 12, and 15, what is the value of x?

SOLUTION: You are asked to determine the LCM of 9, 12, and 15. First, break each number down into its prime factorization:

Since x is a multiple of 9, x must have two 3s in its prime factorization.

Thus x = 3 × 3 . . . . Since x is a multiple of 12, x must have two 2s and one 3 in its prime factorization. You know that x already has a 3 in its prime factorization, so to make x a multiple of 12, you only need to add two 2s to its prime factors. Thus x = 3 × 3 × 2 × 2 . . . .

Since x is a multiple of 15, it must have one 3 and one 5 in its prime factorization. You know from earlier that x already has a 3 in its prime factorization, so to make x a multiple of 15, you only need to add one 5 to its prime factors. Thus x = 3 × 3 × 2 × 2 × 5 = 180.