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Least common multiple
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Least common multiple

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. If there is no such positive integer, e.g., if a = 0 or b = 0, 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

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

If a and b are not both zero, the least common multiple can be computed by using the greatest common divisor (gcd) of a and b:

Thus, the Euclidean algorithm for the gcd also gives us a fast algorithm for the lcm. To return to the example above,

Efficient calculation

The formula

is adequate to calculate the lcm for small numbers using the formula as written.

Because that (ab)/c = a(b/c) = (a/c)b, one can calculate the lcm using the above formula more efficiently, by firstly exploiting the fact that b/c or a/c may be easier to calculate than the quotient of the product ab and c. This can be true whether the calculations are performed by a human, or a computer, which may have storage requirements on the variables a, b, c, where the limits may be 4 byte storage - calculating ab may cause an overflow, if storage space is not allocated properly.

Using this, we can then calculate the lcm by either using:

or
Done this way, the previous example becomes:

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