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) 
See also Greatest Common Divisor, Mangoldt Function, Relatively Prime
References
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: SpringerVerlag, pp. 200201, 1994.
© 19969 Eric W. Weisstein
19990526