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# What are all the factors, the prime factorization, and factor pairs of 108?

To find the factors of 108, divide 108 by each number starting with 1 and working up to 108

Now let us find how to calculate all the factors of 108:
2
= 54
3
= 36
4
= 27
6
= 18
9
= 12
As you can see, the factors of 108 are:
1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 108

## Introduction to Factors

Factors are the building blocks of numbers that, when multiplied together, result in another number. With the illustrative example of 108, we will explore the integers that can be multiplied to reach this specific number. Understanding the factors of 108 provides insight into how this number interacts and relates to other numbers within the realm of mathematics.

## What are Factors of 108?

The concept of factors pertains to numbers that divide 108 without leaving any remainder. When paired and multiplied, these factors will equate to 108. Here’s a visual representation and list of the factors of 108:

• 1 x 108 = 108
• 2 x 54 = 108
• 3 x 36 = 108
• 4 x 27 = 108
• 6 x 18 = 108
• 9 x 12 = 108
• 12 x 9 = 108
• 18 x 6 = 108
• 27 x 4 = 108
• 36 x 3 = 108
• 54 x 2 = 108
• 108 x 1 = 108

## How to Find the Factors of 108?

To determine the factors of 108, one can utilize the division method, which involves dividing 108 by various numbers to discover integer results. Follow this process:

Begin with the smallest number, 1, and divide 108 by each subsequent integer. If the division results in a whole number, then the divisor and the quotient are factors of 108. Continue this process with 108 being divided by all numbers up to itself, as it is a factor of itself.

Here is an example of the division method to find factors of 108:

• 108 ÷ 1 = 108, thus 1 and 108 are factors
• 108 ÷ 2 = 54, thus 2 and 54 are factors
• 108 ÷ 3 = 36, thus 3 and 36 are factors

Note: Any division of 108 by numbers other than its factors will result in a non-integer quotient.

## Pair Factors of 108

Pair factors of 108 refer to sets of two integers that, when multiplied together, result in 108. It is crucial to include both positive and negative factors since the product of two negative integers is also a positive number. Here is a list of both positive and negative pair factors for 108:

Positive Pair Factors Negative Pair Factors
(1, 108) (-1, -108)
(2, 54) (-2, -54)

## Prime Factorization of 108

The prime factorization of 108 involves breaking it down into its prime number components. This process is demonstrated in a factor tree, providing a visual method for finding all prime factors.

Here’s how you can perform prime factorization of 108:

108 ÷ 2 = 54 (2 is a prime factor)

54 ÷ 2 = 27 (2 is a prime factor)

27 ÷ 3 = 9 (3 is a prime factor)

9 ÷ 3 = 3 (3 is a prime factor)

3 ÷ 3 = 1 (3 is a prime factor)

In the end, the prime factors of 108 are 2 and 3, making the prime factorization 2 x 2 x 3 x 3 x 3 or 2^2 x 3^3.

## Important Points to Remember

• The number 108 is a composite number with multiple factors.
• The prime factorization of 108 is 2^2 x 3^3.
• There are both positive and negative pair factors.
• The sum of all factors of 108 is a specific number that can be determined by adding each factor.

## Exercises

• What are the factors of 108?
• How many prime factors does 108 have?
• Calculate the sum of all factors of 108.

## Solutions to Exercises

1. The factors of 108 are 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, and 108.
2. 108 has two distinct prime factors: 2 and 3.
3. The sum of all factors of 108 is 280.