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Least common multiple

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.



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