Least Common Multiple

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In arithmetic and number theory, the least common multiple or lowest common multiple (lcm) or smallest common multiple of two integers a and b is the smallest positive integer that is a multiple of both a and b. Since it is a multiple, it can be divided by a and b without a remainder. If there is no such positive integer, e.g., if a = 0 or b = 0, then lcm(a, b) is defined to be zero.

For example, the least common multiple of the numbers 4 and 6 is 12. We could easily write out a list of multiples for both 4 and 6, like:

4 8 12 16 20 24 28 32
6 12 18 24 30 36 42 48

and determine that there are several multiples that 4 and 6 have in common, but that 12 is the lowest of those. The other method worth discussing would be to write out the prime factors of each of the two numbers, and take the greatest number of prime factors from each. In this example, the prime factors of 4 are 2 · 2 and the prime factors of 6 are 2 · 3. Since 4 has 22 and 6 only has one 2, we would take the greater number of twos (the 22) times the only 3, giving

2^2 \cdot 3 = 12.

Uses and Applications

When adding or subtracting fractions, it is useful to find the least common multiple of the denominators, often called the lowest common denominator. For instance,

{2\over21}+{1\over6}={4\over42}+{7\over42}={11\over42},

where the denominator 42 was used because lcm(21, 6) = 42.

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