Least common multiple calculator

The least common multiple is the smallest number that is a multiple of every value you enter.

It is used to add fractions with different denominators and to work out how often two repeating cycles line up.

It is derived from the greatest common divisor: the product of two numbers divided by their GCD gives the LCM. With more than two values the same operation is applied cumulatively.

Adding fractions requires a common denominator

A third plus a quarter is not two sevenths. Both must be taken to the same denominator, and the lowest common multiple of three and four is twelve: four twelfths plus three twelfths is seven twelfths. Any common multiple works, but the lowest one saves simplifying afterwards.

Gears and cycles that coincide

Two gear wheels of twelve and eighteen teeth return to their starting position every thirty-six teeth, their lowest common multiple. The same idea explains when two buses running every fifteen and every twenty minutes coincide, or when two planets realign.

Relationship with the greatest common divisor

The product of two numbers always equals the product of their greatest common divisor and their lowest common multiple. That identity lets you calculate one from the other without repeating the factorisation, and it is the basis of the fast method.

Calendars and leap years

The twenty-eight year cycle after which the calendar repeats exactly comes from combining the seven days of the week with the four-year leap cycle: their lowest common multiple is twenty-eight.