## Least Common Multiple (LCM)

The Least Common Multiple (LCM) or Lowest Common Multiple (LCM) of two integers is the smallest positive integer that is a multiple of both numbers. Since it is a multiple, each integer divides it without remainder. If there is no such positive integer, then LCM between the two integers is defined as zero.

Consider the LCM of the integers 2 and 3.

Multiples of 2: 2, 4, 6, 8, 10, 12, 14, 16, 18, ...

Mulitples of 3: 3, 6, 9, 12, 15, 18, ...

The shared multiples between both integers consist of 6, 12, 18, ... Since the smallest common multiple is 6, it follows that LCM(2, 3) = 6

While the LCM for small numbers may be determined by inspection, the method for finding the LCM for large numbers consists of the following:

- Write each number as a product of prime numbers
- LCM may be attained by multiplying common factors of both numbers; Each common factor should be selected according to the greatest number of times it exists

Example #1:

Find LCM(12, 80)

12 = 2 x 2 x 3

80 = 2 x 2 x 2 x 2 x 5

The number 2 is a factor of both numbers, however, as it occurs four times for 80, those factors are selected

The numbers 3 is only common to 12, therefore, it should be included as a factor

The number 5 is only common to 80, therefore, it should be included as a factor

LCM(12, 80) = 2 x 2 x 2 x 2 x 3 x 5

LCM (12, 80) = 240

Find LCM(12, 30, 48)

Solution #1:

12 = 2 x 2 x 3

80 = 2 x 2 x 2 x 2 x 5

The number 2 is a factor of both numbers, however, as it occurs four times for 80, those factors are selected

The numbers 3 is only common to 12, therefore, it should be included as a factor

The number 5 is only common to 80, therefore, it should be included as a factor

LCM(12, 80) = 2 x 2 x 2 x 2 x 3 x 5

LCM (12, 80) = 240

Example #2:

Find LCM(12, 30, 48)

Solution #2: