Least common multiple (original) (raw)
The least common multiple (LCM) of two integers a and b is the smallest positive integer that is a multiple of both a and b. If there is no such positive integer, i.e., if either a or b is zero, then lcm(a,b) is defined to be zero.
The least common multiple is useful when adding or subtracting fractions, because it yields the lowest common denominator. Consider for instance
2/21 + 1/6 = 4/42 + 7/42 = 11/42
the denominator 42 was used because lcm(21,6) = 42.
In case not both a and b are zero, the least common multiple can be computed by using the greatest common divisor (or GCD) of a and b,
a b | |
---|---|
lcm(a, b) = | --------- |
gcd(a, b) |
Thus, the Euclidean algorithm for the GCD also gives us a fast algorithm for the LCM. As an example, the LCM of 12 and 15 is 12 × 15 / 3 = 60.