## Least Common Multiple

The least common multiple of two numbers and is denoted or and can be obtained by finding the Prime factorization of each

 (1)

 (2)

where the s are all Prime Factors of and , and if does not occur in one factorization, then the corresponding exponent is 0. The least common multiple is then
 (3)

Let be a common multiple of and so that
 (4)

Write and , where and are Relatively Prime by definition of the Greatest Common Divisor . Then , and from the Division Lemma (given that is Divisible by and ), we have is Divisible by , so
 (5)

 (6)

The smallest is given by ,
 (7)

so
 (8)

 (9)

The LCM is Idempotent
 (10)

Commutative
 (11)

Associative
 (12)

Distributive
 (13)

and satisfies the Absorption Law
 (14)

It is also true that
 (15)

Guy, R. K. Density of a Sequence with L.C.M. of Each Pair Less than .'' §E2 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 200-201, 1994.