least common multiple

The least common multiple of two numbers is the smallest whole number that both of them divide into evenly. The multiples of 4 are 4, 8, 12, 16, 20, and the multiples of 6 are 6, 12, 18, 24; the first number on both lists is 12, so the least common multiple of 4 and 6 is 12.

Listing multiples until the lists meet is the safest method for small numbers. For larger ones, factor each number into primes, then multiply each prime the greatest number of times it appears in any single list: 8 is 2 × 2 × 2 and 12 is 2 × 2 × 3, so the least common multiple is 2 × 2 × 2 × 3 = 24. Multiplying the two numbers together always gives a common multiple, but a wrong answer can come from that shortcut when the numbers share a factor.

Where it is used: adding or subtracting fractions with unlike denominators, since the least common multiple of the denominators is the smallest common denominator that works. It is the mirror of the greatest common factor — the smallest number both divide into, against the largest number that divides both.