# Least Common Multiple (L.C.M.)

In the last topic we have discussed about common multiples that is same multiples of two or more numbers. Now here we will discuss about the least common multiple that is the smallest common multiple of two or more numbers.

Here are few examples illustrated on least common multiples:

1. Find the least common factor of 12 and 18

Solution:

The multiples of 12 are: 12, 24, 36, 48, 60, 72, 84 …

The multiples of 18 are: 18, 36, 54, 72 …

The common multiples of 12 and 18 are: 36, 72 …

The least common multiple of 12 and 18 is 36

This problem can be solved by another method as well that is by complete factorization. The product of highest power of all the factors will be the least common multiple i.e. L.C.M

Alternatively,

The prime factors of 12 are: 2 × 2 × 3 = 22 × 3

The prime factors of 18 are: 2 × 3 × 3 = 32 × 2

The highest power of 2 is 22

The highest power of 3 is 32

Therefore, least common multiple of 12 and 18 is 2 × 2 × 3 × 3 = 22  × 3= 36

2. Find the least common factor of 15 and 25

Solution:

The multiples of 15 are: 15, 30, 45, 60, 75, 90, 105, 120, 135, 150, 165 …

The multiples of 25 are: 25, 50, 75, 100, 125, 150, 175 …

The common multiples of 15 and 25 are: 75, 150 ….

The least common multiple of 15 and 25 is 75

3. Find the least common factor of 8 and 16

Solution:

The multiples of 8 are: 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96 …

The multiples of 16 are: 16, 32, 48, 64, 80, 96 …

The common multiples of 8 and 16 are: 16, 48, 64, 80, 96 ….

The least common multiple of 8 and 16 is 16

4. Find the least common factor of 20 and 30

Solution:

The multiples of 20 are: 20, 40, 60, 80, 100, 120, 140, 160, 180, 200, 220, 240, 260 …

The multiples of 30 are: 30, 60, 90, 120, 150, 180, 210, 240, 270, 300 …

The common multiples of 20 and 30 are: 60, 120, 180, 240 …

The least common multiple of 20 and 30 is 60

5. Find the least common factor of 9 and 6

Solution:

The multiples of 9 are: 9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99, 108 …

The multiples of 6 are: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72 ….

The common multiples of 9 and 6 are: 18, 36, 54, 72 …

The least common multiple of 9 and 6 is: 18

5. Find the least common factor of 21 and 14

Solution:

The multiples of 21 are: 21, 42, 63, 84, 105, 126, 147, 168, 189, 210 ….

The multiples of 14 are: 14, 28, 42, 56, 70, 84, 98, 112, 126, 140 …

The common multiples of 21 and 14 are: 42, 84, 126 …

The least common multiple of 21 and 14 is 42