Abstract Algebra: Theory and Applications · Problem 2.4.23

Groups — Thomas Judson

Mathematics — Abstract Algebra · least-common-multiple

Problem

Define the least common multiple of two nonzero integers

a

and

b

, denoted by

\text{lcm}(a,b)

, to be the nonnegative integer

m

such that both

a

and

b

divide

m

, and if

a

and

b

divide any other integer

n

, then

m

also divides

n

. Prove there exists a unique least common multiple for any two integers

a

and

b

. Hint. Use the Principle of Well-Ordering and the division algorithm.