# 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

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*:

## Efficient calculation

The formula

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:

## External links