# How to Find Lowest Common Multiple (LCM) of Expressions?

How to find the lowest common multiple (LCM) of two or more expressions in maths? Detailed solutions to examples are presented and full solutions to the questions in Detailed Solutions and explanations are included.

## What is the lowest common multiple (LCM) of 2 or more maths expressions?

The lowest common multiple of two or more expressions is the smallest (or simplest) expression that is divisible by each of these expressions. It is found by first
factoring completely each of the given expressions then use these factors to write the LCM. Detailed examples are shown below.

### Example 1

Find the lowest common multiple of the expressions x 2 - 1 and x - 1.
Solution
We first factor the given expressions
x 2 - 1 = (x - 1)(x + 1)
x - 1 = x - 1
We now make the LCM by multiplying all factors included in the factoring of the given expressions. Common factors are used once only and the one with the highest power is used.
x - 1 is a factor to both expression and will therefore be used once. x + 1 is a factor in the first expression and will therefore be used. Hence
LCM ( x 2 - 1 and x - 1 ) = (x - 1)(x+1)

### Example 2

Find the lowest common multiple of the expressions 2 x 2, x 2 + x and x 3 + 2 x .
Solution
We first factor the given expressions completely:
2 x 2 = 2 x 2
x 2 + x = x(x + 1)
x 3 + 2 x = x( x 2 + 2)
The LCM is made by multiplying all factors included in the factoring of the given expressions. Common factors are used once only and the one with the highest power is used.
2 is a factor in the first term only and will therefore be used. x is a factor in all three expressions and the one with the highest power which is x 2 in the first term is used. x + 1 is a factor in the second expression only and is therefore used. x 2 + 1 is a factor in the third expression only and is therefore used. Hence
LCM ( 2x 2, x 2 + x , x 3 + 2 x ) = 2x 2 (x + 1) (x 2 + 2)

### Example 3

Find the lowest common multiple of the expressions x 2 + 3 x - 4, (x - 1) 2 and x 2 + 9 x + 20.
Solution
We first factor the given expressions completely:
x 2 + 3 x - 4 = (x - 1)(x + 4)
(x - 1) 2 = (x - 1) 2
x 2 + 9 x + 20 = (x + 4)(x + 5)
The LCM is made by multiplying all factors included in the factoring of the given expressions. Common factors are used once only and the one with the highest power is used.
x - 1 is a factor in the first and second expressions is therefore the one with the highest power (x - 1) 2 in the second expression is used. x + 4 is a factor in the first and third expressions is used once only. x + 5 is a factor in the third expression only and is therefore used . Hence
LCM ( x 2 + 3 x - 4 , (x - 1) 2 , x 2 + 9 x + 20 ) = (x - 1) 2 (x + 4)(x + 5)

Find Lowest Common Multiple (LCM) of the expressions given below.
1. 2 (x + 1) and 3 (x + 1) .
2. 2 (x - 1) 2 and 5 (x - 1) .
3. x 2 + 5 x + 6 and 2 x 2 + 2 x - 4 .
4. 3 x 3 - 2 x 2 - x and x - 1 .
5. 3 x 3 - 2 x 2 - x , 2 x 2 - 2 and (x - 1) 2.

Detailed Solutions and explanations to the above questions are included.