Let us consider the list of multiples of 2, 3, and 4.
Multiples of 2: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, ...
Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, ...
Multiples of 4: 4, 8, 12, 16, 20, 24, ...
The common multiples are 12, 24, ...
But the Least Common Multiple (LCM) is 12.
Number 12 is the smallest number divisible by 2, 3, and 4.
The Least Common Multiple (LCM) of a set of counting numbers is the smallest number divisible by each number in the set.
Example: Find the LCM of 20, 30 and 40
Multiples of 20: 20, 40, 60, 80, 100, 120, ...
Multiples of 30: 30, 60, 90, 120, 150, 180, ...
Multiples of 40: 40, 80, 120, 160, 200, ...
LCM is 120.
Another way of finding the LCM of two or more numbers is by using prime factorization.
20 = 2 x 2 x 5 (two 2's, one 5)
30 = 2 x 3 x 5 (one per number)
40 = 2 x 2 x 2 x 5 (three 2's one 5)
LCM is 2 x 2 x 2 x 3 x 5 = 120
The LCM is the product of the primes with the most number of times it appears in any one of the prime factorization and other prime numbers.
Another example, the LCM of 12, 15, 30 using prime factorization:
12 = 2 x 2 x 3
15 = 3 x 5
30 = 2 x 3 x 5
Since there are two 2's in 12 then multiply it with other prime numbers you see on the list,
LCM is 2 x 2 x 3 x 5 = 60.
