Least common multiple
The least common multiple
(LCM) of two integers a
is the smallest positive integer that is a multiple of both a
. If there is no such positive integer, i.e., if either a
) 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) = ||---------|
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.