What is the Least Common Multiple? Definition and Examples


What is a least common multiple? A least common multiple (LCM) is the smallest common multiple that two or more numbers share or have in common. For example, 6 is the least common multiple of 2 and 3.

Examples showing how to find the least common multiple of two or three numbers


Example #1

Find the least common multiple of 2 and 3.

First, list the multiples of each number. In our example here, we will show only the first 15 multiples of each number.

Multiples of 2: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, ...

Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, ...

As shown in red, the first four common multiples of 2 and 3 are 6, 12, 18 and 24.

As you can clearly see, the smallest common multiple is 6. The LCM of 2 and 3 is 6

Example #2

Find the least common multiple of 2, 4 and 5.

First, list the multiples of each number. In our example here, we will show only the first 20 multiples of each number.

Multiples of 2: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, ...

Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64, 68, 72, 76, 80, ...

Multiples of 5: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 100, ...

As shown in red, the first two common multiples of 2, 4, and 5 are 20, and 40.

As you can clearly see, the smallest common multiple is 20. The LCM of 2, 4 and 5 is 20.

least common multiple of prime numbers

Suppose n and m represent two distinct prime numbers. Then, the least common multiple of n and m is n × m.

Example #3

Find the least common multiple of 3 and 5.

Since 3 and 5 are prime numbers, the least common multiple of 3 and 5 is 3 × 5 = 15

Example #4

Find the least common multiple of 7 and 11.

Since 7 and 11 are prime numbers, the least common multiple of 7 and 11 is 7 × 11 = 77

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