# Definition of Least Common Multiple (LCM)

**Definition:**

A multiple of a number is the product of the number and another whole number. Factors and multiples are closely related.

For example, 9 is a factor of 90, so 90 is a multiple of 9.

**Denition: **

The least common multiple or lcm of two or more numbers is the lowest number that is a multiple of each of the numbers. One method of nding the least common multiple involves listing the multiples of each number until you nd the rst number that is a multiple of both.

**Example Find the least common multiple of 6 and 8.**

The first few multiples of 6 are 6 12 18 24 30 36 42 48…

The first few multiples of 8 are 8 16 24 32 40 48 56 64…

The first multiple that occurs in both is 24, so the least common multiple is 24. This method may take too long if we are dealing with larger numbers. Instead, we will use another method involving prime factors.

**Example:**

*Find the least common multiple of 108 and 180.*

The least common multiple must include all of the prime factors that occur in each of the numbers. * Therefore we need two 2s, three 3s and one 5*.

Therefore, the least common multiple is * 2 x 2 x 3 x 3 x 3 x 5 = 540*.

Notice that by looking at the prime factors we can tell that 540 = 108 x 5 and

540 = 180 x 2 x 3.

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