Finding the number of factors of any number

If a number can be expressed as a product of two whole numbers, then the whole numbers are called factors of that number.

In other words, A factor is a whole number which divides exactly into a whole number, leaving no remainder.

For example, 13 is a factor of 52 because 13 divides exactly into 52 (52 ÷ 13 = 4 leaving no remainder).

52 = 1 x 52 = 2 x 26 = 4 x 13

So, the complete list of factors of 52 is: 1, 2, 4, 13, 26, and 52 (all these divide exactly into 52).

A simple technique to find the number of factors of a given number is to express the number as a product of powers of prime numbers or prime factors.

To illustrate let’s find the numbers of factors of our example 52.

Note that, 52 can be expressed as 4 x 13 = 2² x 13

So, the prime factors of 52 are 2 and 13.

Now, increment the power of each of the prime numbers by 1 and multiply the result.

In this case it will be (2 + 1) x (1 + 1) = 3 x 2 = 6 (power of 2 is 2 and power of 13 is 1)

Therefore, there will 6 factors including 1 and 52.

Also note that, all numbers have a factor of 1 since 1 multiplied by any number equals that number. All numbers can be divided by themselves to produce the number 1. Therefore, we normally ignore 1 and the number itself as useful factors.

So, excluding, these two numbers, you will have (6 – 2) = 4 factors.

To be certain the factors are: 2, 4, 13 and 26.

To further illustrate let’s find the numbers of factors of 48.

48 can be written as 16 x 3 = 2⁴ x 3

So, the prime factors of 48 are 2 and 3.

Now, increment the power of each of the prime numbers by 1 and multiply the result.

In this case it will be (4 + 1) x (1 + 1) = 5 x 2 = 10 (the power of 2 is 4 and the power of 3 is 1)

Therefore, there will be 10 factors including 1 and 48. Excluding, these two numbers, you will have (10 – 2) = 8 factors. And the factors are: 2, 3, 4, 6, 8, 12, 16 and 24

Another example: 216 = 8 x 27 = 2³ x 3³

Here also, the prime factors of 216 are 2 and 3.

Now, increment the power of each of the prime numbers by 1 and multiply the result.

(3+1) x (3+1) = 4 x 4 = 16 (power of both 2 and 3 is 3)

Therefore, there will be 16 factors including 1 and 216.

Excluding, these two numbers, you will have 16 – 2 = 14 factors which are 2, 3, 4, 6, 8, 9, 12, 18, 24, 27, 36, 54, 72 and 108.