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:
12 = 2 x 2 x 3
30 = 2 x 3 x 5
48 = 2 x 2 x 2 x 2 x 3
The number 2 is a factor of all three numbers, however, as it occurs four times for 48, those factors are selected
The number 3 is a factor of both 12 and 48 a single time, therefore, it should be included as a factor
The number 5 is only common to 30, therefore, it should be included as a factor
LCM(12, 30, 48) = 2 x 2 x 2 x 2 x 3 x 5
LCM(12, 30, 48) = 240
30 = 2 x 3 x 5
48 = 2 x 2 x 2 x 2 x 3
The number 2 is a factor of all three numbers, however, as it occurs four times for 48, those factors are selected
The number 3 is a factor of both 12 and 48 a single time, therefore, it should be included as a factor
The number 5 is only common to 30, therefore, it should be included as a factor
LCM(12, 30, 48) = 2 x 2 x 2 x 2 x 3 x 5
LCM(12, 30, 48) = 240
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