Editing Least common multiple (section)

=== Using the greatest common divisor ===
The least common multiple can be computed from the [[greatest common divisor]] (gcd) with the formula
:<math>\operatorname{lcm}(a,b)=\frac{|ab|}{\gcd(a,b)}.</math>

To avoid introducing integers that are larger than the result, it is convenient to use the equivalent formulas
:<math>\operatorname{lcm}(a,b)=|a|\,\frac{|b|}{\gcd(a,b)} = |b|\,\frac{|a|}{\gcd(a,b)} ,</math>
where the result of the division is always an integer.

These formulas are also valid when exactly one of {{math|''a''}} and {{math|''b''}} is {{math|0}}, since {{math|1=gcd(''a'', 0) = {{abs|''a''}}}}. However, if both {{math|''a''}}{{math|}} and {{math|''b''}} are {{math|0}}, these formulas would cause [[division by zero]]; so, {{math|1=lcm(0, 0) = 0}} must be considered as a special case.

To return to the example above,

:<math>\operatorname{lcm}(21,6)
=6\times\frac {21}{\gcd(21,6)}
=6\times\frac {21} 3
=6\times 7
=  42.
</math>

There are fast [[algorithm]]s, such as the [[Euclidean algorithm]] for computing the gcd that do not require the numbers to be [[Integer factorization|factored]]. For very large integers, there are even faster algorithms for the three involved operations (multiplication, gcd, and division); see [[Fast multiplication]]. As these algorithms are more efficient with factors of similar size, it is more efficient to divide the largest argument of the lcm by the gcd of the arguments, as in the example above.