Divisors, prime factors, multiples

Divisors and multiples

Divisors (also called factors) and multiples are essential concepts for the GRE.

A divisor is a number that divides another number evenly (with no remainder). For example, $3$ is a divisor of $12$ because $12$ can be divided by $3$. However, $3$ is not a divisor of $13$ because $13$ can’t be divided evenly by $3$.

A multiple is the result of multiplying a number by an integer. 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 ($12÷12=1$), 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. $36$

Try it before you look at the solution.

Prime factors

A prime factor (also called a prime divisor) is a factor of a number that is also prime.

If a question asks for the prime factors of a number, you list the prime numbers that divide evenly into the original number.

A simple way to find prime factors is to use a factor tree. Here’s an example of the number 12’s prime factor tree:

To use this method, split the original number into two divisors. Then keep splitting each divisor into smaller divisors until every branch ends in prime numbers. The order you split the numbers doesn’t matter; you’ll end with the same prime numbers.

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 written as $2^2\times 3$.

Factoring shortcuts

A quick way to factor a number is to test small prime numbers (i.e. $2,3,5,7,11,13,17,19,23,...$) in ascending order and see which ones divide evenly.

• If a prime divides evenly, it’s a factor.
• A prime can be a factor more than once, so keep dividing by it until it no longer divides evenly.
• Then move to the next prime.

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:

• $1^2=1$
• $2^2=4$
• $3^2=9$
• $4^2=16$
• $5^2=25$
• $6^2=36$
• $7^2=49$
• $8^2=64$
• $9^2=81$
• $10^2=100$
• $11^2=121$
• $12^2=144$

Small cubes:

• $1^3=1$
• $2^3=8$
• $3^3=27$
• $4^3=64$
• $5^3=125$
• $6^3=216$
• $7^3=343$
• $8^3=512$
• $9^3=729$
• $10^3=1000$
• $11^3=1331$
• $12^3=1728$

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: $13x\times 11y=1430$

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

Try to solve it before continuing.

The key is to factor 1,430. The equation $13x\times 11y=1430$ tells you that 1,430 must be divisible by both 13 and 11.

Start with a factor tree:

Now you’re left with 10 to factor. The prime factors of 10 will be the remaining values for $x$ and $y$:

The question says $x$ and $y$ are both greater than 1, and $x>y$. That means $x=5$ and $y=2$. Then

$\frac{x}{y}=\frac{5}{2}=2.5$.

Remainders

When you divide a number by another number that is not a divisor (i.e. not a factor) of that number, the result won’t be a whole number. For example, $7/2=3.5$.

You can also write this as division with a remainder: $7/2$ is 3 with a remainder of 1. In other words, 7 can be split into two groups of 3, with 1 left over.

To find the remainder from a decimal result, multiply the decimal part by the divisor. In this case, the decimal part is $0.5$, so $0.5\times 2=1$. That 1 is the remainder.

What is the remainder of $142/12$? A calculator gives about $11.833$. Multiply the decimal part by the divisor: $0.833\times 12\approx 10$. So $142/12=11$ remainder $10$.

Note: Your calculator may show $9.999$ instead of $10$ because $0.833$ is a rounded version of a repeating decimal. This is a small rounding error you can 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

Break it down step by step.

Quantity A:

• $198/12=16.5$
• The decimal part is $0.5$, so the remainder is $0.5\times 12=6$
• Twice the remainder is $6\times 2=12$

Quantity B:

• The remainder when dividing by 4 must be one of $[0,1,2,3]$
• Three times that remainder must be at most $3\times 3=9$

So Quantity B can’t exceed 9, while Quantity A is 12. Therefore, Quantity A is greater.

For practice, you can also compute Quantity B directly:

• $175/4=43.75$
• The decimal part is $0.75$, so the remainder is $0.75\times 4=3$
• Three times the remainder is $3\times 3=9$

This matches the 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: