What is a Divisor?

A divisor is a number that divides another number in a division equation. It doesn’t matter whether the number divides it completely or produces a remainder. However, when someone asks you the divisors of a certain number, they are usually referring to factors. Both words are used interchangeably, although they have a slight difference. Factors are the numbers that divide a number completely without a remainder.  Factors can also be defined as all the divisors of a number that do not produce a remainder. Not all divisors are factors of a number, but all factors of a number are divisors of that number.

For example

20   ÷ 4 = 5

In the example above, 4 is a divisor of 20. If we replace 4 with 3, it will be a divisor but with a remainder. However, 4 is a factor of 20 while 3 is not. Let’s look at different ways to find the divisors of a number.

How to find divisors of a number?

There are several ways to find the divisors of a number. The first option is to divide it by every possible number which is less or equal. But this can be an exhaustive process, especially if the number is very large. If you are working with a small number, then you can use the exhaustive division method.

Method 1: Using the division Method

Start by dividing the number by itself until and by all the numbers less than it up to 1.

Next, List only the numbers that did not result in a decimal or remainder and ignore the rest

For example, you can find the divisors of 8 by dividing it with every number less or equal to it.

8 ÷ 8 = 1

8 ÷ 7 = 1.1429

8 ÷ 6 = 1.3333

8 ÷ 5 = 1.6

8 ÷ 4 = 2

8 ÷ 3 = 2.6667

8 ÷ 2 = 4

8 ÷ 1 = 8

Therefore the divisors of 8 are 1, 2, 4, and 8

Method 2: Prime Factorization  method

In this method, you need to list the prime factors of the number first.

For Example, Let’s find the divisors of 24 using the decomposing method

2 * 2 * 2 * 3

We already know two divisors which are 1 and 24

From the above list of prime factors, we 2 and 3 are our next divisors.

So far, the divisors are 1, 2, 3, and 24

Now find every possible product from the prime factors of 24

2 * 2 = 4

2* 3 = 6

2* 2 * 2 = 8

2* 2 *3 = 12

Divisor Facts

• The divisor can divide a number completely (Without a decimal) or partially (With a decimal).
• A divisor can be a negative or positive number
• The divisor is always larger than the remainder
• Both -1 and 1 can divide every number
• If there is no remainder, then it means the divisor has divided the dividend completely
• A divisor can also be a number that divides a number without resulting in a remainder.
• When the divisor and the dividend are equal, then the quotient will always be 1. For example, 50 ÷ 50 = 1
• When the dividend is equal to the quotient, then the divisor is 1. For example 50 ÷ 1 = 50

Different ways to represent the divisor

How to find the total number of Divisors

d(n) = (a+1) (b+1) (C+1)

Where a, b, c are exponents and n the number (d stands for divisor in this equation)

Example 1: Find the sum of divisors of 72

Step 1: Find the exponents of 72 by breaking it down to its prime factorization

2 × 2 × 2 × 3 × 3 = 2³ × 3²

The exponents of 72 are 3 and 2

Step 2: Apply the exponents to the equation

d(n) = (a+1) (b+1) (C+1)

d(72) = (3+1) (2+1)

Step 3: Simplify the Equation

d(72) = (4) (3)

d(72) = 12

Therefore the total number of divisors for 72 is 1