Divisors, prime factors, multiples | Arithmetic & algebra | Quantitative reasoning

Divisors and multiples

Divisors (a.k.a. factors) and multiples are essential concepts for the GRE. A divisor is a number that you can divide into another number. For example, 3 is a divisor of 12, because 12 can be divided by 3 evenly with no fraction remaining. However, 3 is not a divisor of 13 because 13 cannot be split evenly in 3 ways. A multiple can be considered the opposite of a divisor. A multiple of 12 is any product of 12 and another number. For example, 24 is a multiple of 12, because $2 \times 12 = 24$ .

Any number is both a divisor of itself and a multiple of itself.

For instance, 12 can be split evenly into 12 parts ( $12 \div 12 = 1$ ), and 12 is the product of 12 and 1 ( $12 \times 1 = 12$ ).

Let’s try an example.

Which of the following numbers is a divisor of 36 and a multiple of 4?

Select all that apply.
A. 3
B. 4
C. 8
D. 9
E. 12
F. 18
G. 36

Have you given it a try?

(spoiler)

First, determine the factors (divisors) of 36:

1 and 36 are freebies: $36/1 = 36$
2 and 18: $36/2 = 18$
3 and 12: $36/3 = 12$
4 and 9: $36/4 = 9$
6 (and 6): $36/6 = 6$

So, here are all the factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, and 36.

Of these, 4, 12, and 36 are also a multiple of 4.

Prime factors

A prime factor (a.k.a. prime divisor) is a number that can be divided into a larger number and is also prime. If a question asks for the prime factors of a number, you need to list all the prime numbers that can be divided evenly into the original number. The simplest way to find the prime factors of any number is to use a factor tree. Here’s an example of the number 12’s prime factor tree:

Factor free of 12

To use this method, simply split the original number into two divisors. Continue to split those two divisors into their own smaller divisors until every branch ends with prime numbers. The order you use to split the numbers doesn’t matter; you’ll always end up with the same prime numbers at the end.

The unique prime factors of $12$ are $2$ and $3$ . The prime factorization of $12$ is $2 \times 2 \times 3$ , which can also be expressed as $2^{2} \times 3$ .

Factoring shortcuts

The quickest method to factor a number is often to simply use your calculator and go through small prime numbers (i.e. $2, 3, 5, 7, 11, 13, 17, 19, 23, ...$ ) in ascending order to see which ones divide evenly. If a number divides evenly, it is a factor. A number might be a factor multiple times, so continue dividing the result by it again until it doesn’t divide evenly, and then move up to the next prime number.

For example, here are steps for factoring $462$ :

Factors so far: (none), value remaining: $462$
- Calculator: $462/2 = 231$ with no decimal; this is a factor
Factors so far: $[2]$ , value remaining: $231$
- Trying $2$ again since it could be a factor multiple times
- Calculator: $231/2 \approx 115.5$ with a decimal; try next prime
- Calculator: $231/3 = 77$ with no decimal; this is a factor
Factors so far: $[2, 3]$ , value remaining: $77$
- Trying $3$ again…
- Calculator: $77/3 \approx 25.7$ with a decimal; try next prime
- Calculator: $77/5 \approx 15.4$ with a decimal; try next prime
- Calculator: $77/7 = 11$ with no decimal; this is a factor
Factors so far: $[2, 3, 7]$ , value remaining: $11$
- Prime check: $11$ is prime; we’re done
Prime factorization of $462 : [2, 3, 7, 11]$

Sidenote

Memorize these tidbits to save time during your exam

Small prime numbers: $2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47$

Small squares:

$1^{2} = 1$
$2^{2} = 4$
$3^{2} = 9$
$4^{2} = 16$
$5^{2} = 25$
$6^{2} = 36$
$7^{2} = 49$
$8^{2} = 64$
$9^{2} = 81$
$1 0^{2} = 100$
$1 1^{2} = 121$
$1 2^{2} = 144$

Small cubes:

$1^{3} = 1$
$2^{3} = 8$
$3^{3} = 27$
$4^{3} = 64$
$5^{3} = 125$
$6^{3} = 216$
$7^{3} = 343$
$8^{3} = 512$
$9^{3} = 729$
$1 0^{3} = 1000$
$1 1^{3} = 1331$
$1 2^{3} = 1728$

Small divisibility rules:

A number is divisible by 2 if it is even
A number is divisible by 3 if the sum of the digits is divisible by 3 (e.g. try 222,108 = sum of digits is 2+2+2+1+8=15, which is divisible by 3)
A number is divisible by 4 if the last two digits are divisible by 4 (e.g. try 222,108 = last two digits are 08, which is divisible by 4)
A number is divisible by 5 if the last digit is 0 or 5

Example factoring question

Let’s work through an example factoring question.

Given: $13 x \times 11 y = 1430$

Which of the following is the value of $x / y$ if both variables are greater than $1$ and $x$ is greater than $y$ ?

A. 0.1
B. 0.2
C. 2.5
D. 5
E. 10

Take a moment and try to solve it yourself before continuing!

(spoiler)

Answer: C. 2.5

The key to this problem is to factor 1,430. The question gives us a considerable hint, as it essentially tells us that 1,430 must be divisible by 13 and 11, since they are both part of the numbers that must be multiplied together to reach 1,430 (i.e. $13 * x * 11 * y = 1430$ ). So let’s start with a factor tree:

Partial prime factorization factor tree of 1430

Now we’re left with just 10 to be factored, and the prime factors of 10 will be our values for $x$ and $y$ :

Partial prime factorization factor tree of 1430

The question states that $x$ and $y$ must be greater than 1, and that $x$ is greater than $y$ . This information lets us determine that $x = 5$ and $y = 2$ . We just need one final step to calculate the value of $x / y$ , which is $5/2 = 2.5$ .

Remainders

When you divide a number by another number that is not a divisor (i.e. not a factor) of that number, the answer will not be a whole number. For example, $7/2 = 3.5$ . Since $7$ is an odd number and is not divisible by $2$ , we’re left with this decimal $0.5$ . Another way of expressing the result of $7/2$ is $3$ with a remainder of $1$ . The number $7$ can be split into two groups of $3$ with an extra $1$ remaining. To find the remainder (instead of finding an answer with a decimal), multiply the decimal by the divisor. In this case, we would multiply the decimal $.5$ by $2$ to get $0.5 * 2 = 1$ . The remainder of $7$ divided by $3$ is $1$ .

What is the remainder of $142/12$ ? Plug this into your calculator and you’ll get an answer of about $11.833$ . If we multiply the decimal $(0.833)$ by the divisor ( $12$ ) we can find the remainder. Since $12 * 0.833 = 10$ , we get $142/12 = 11$ remainder $10$ .

Note: Your calculator only shows that the answer is $9.999$ instead of $10$ because you cannot write the repeating $3$ at the end of $.833$ forever. It’s a tiny rounding error that you can just ignore!

Let’s try another example question:

Compare Quantity A to Quantity B.

Quantity A: Twice the remainder of 198/12
Quantity B: Three times the remainder of 175/4

Think you know the answer?

(spoiler)

Answer: Quantity A is greater

This question becomes simple if we break it down step by step.

Quantity A asks us to divide $198/12 = 16.5$ , and then we can find the remainder by multiplying $0.5 * 12 = 6$ . Twice the remainder is $6 * 2 = 12$ .

If we think about this carefully, we don’t even need to solve for Quantity B. The calculation in Quantity B is based on the remainder of 175 when divided by 4, which means that the remainder must be one of the values $[0, 1, 2, 3]$ . That remainder is then multiplied by 3, so it is guaranteed that the maximum value of Quantity B will be 9. Even without finding the exact value of Quantity B, we’ve already determined that Quantity A is greater.

Nonetheless, let’s solve for Quantity B anyway for practice. Dividing out $175/4 = 43.75$ then multiplying the divisor by the decimal gives us a remainder of $0.75 * 4 = 3$ . The Quantity is three times that: $3 * 3 = 9$ .

This matches our shortcut: Quantity A is 12, while Quantity B is 9.

Bringing it all together: question walkthrough video

Here’s a video going through one of our practice questions to demonstrate these ideas in action: