GRE Quant: Factors and Multiples

Factors of a number are integers that divide the given number, while multiples of a number are integers divisible by the given number. Factors are finite, while multiples are infinite. The GRE tests questions on the number of factors, so due coverage of factors and multiples is essential in an end-to-end GRE prep course. This page brings together all the resources you need to learn this concept well. The video and the theory that follow build complete conceptual understanding, show examples in action, and work through GRE style questions so you experience direct application in exam like scenarios. Use this resource to strengthen both your concepts and your method for solving factors and multiples based questions, then apply your learning in GRE practice exercises, GRE practice sectional tests, and GRE practice full length tests. 

 

Overview: Factors and Multiples

Factors

Factors of a number are integers that can divide the given number evenly. For any specific number, the set of factors is finite, meaning there is a limited number of them.

Multiples

Multiples of a number are integers that are divisible by the given number. For any specific number, the set of multiples is infinite, meaning the list continues forever.

Example: The Number 20

• Factors: 1, 2, 4, 5, 10, 20
• Multiples: 20, 40, 60… (and so on)

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Conceptual Practice Set: Factors and Multiples

Correct Answers: A, B, C, F

 

For a detailed explanation, please refer to the video presented earlier on this page.

 

Following is a concise, stepwise written explanation…

 

Question

p and q are positive integers divisible by 3. Which of the following must be true?

Indicate all such statements.

1. p + q is a multiple of 3
2. p – q is a multiple of 3
3. p × q is a multiple of 9
4. p/q is not a multiple of 3
5. p + q is a multiple of 6
6. p^q is a multiple of 27

 

Explanation

Define the variables

Since p and q are divisible by 3, we can write them as p = 3a and q = 3b, where a and b are positive integers.

A. Test p + q is a multiple of 3

3a + 3b = 3 * (a + b).

This result is always divisible by 3.

This is true.

B. Test p – q is a multiple of 3

3a – 3b = 3 * (a – b).

This result is always divisible by 3.

This is true.

C. Test p x q is a multiple of 9

(3a) * (3b) = 9 * (ab).

This result is always divisible by 9.

This is true.

D. Test p / q is not a multiple of 3

If p = 18 and q = 3, then 18 / 3 = 6.

Since 6 is a multiple of 3, the statement is false.

E. Test p + q is a multiple of 6

If p = 3 and q = 6, then 3 + 6 = 9.

Since 9 is not a multiple of 6, the statement is false.

F. Test p^q is a multiple of 27

Since p is a multiple of 3, then p^q is at least 3³ because the smallest value for q is 3. 3³ = 27.

Therefore, p^q will always be divisible by 27.

This is true.

 

Correct Answers: A, B, C, F

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