How I prove the uniqueness of LCM of two non zero integers a, b?

Question

Prove that the least common multiple of two non zero integers is unique. Need to know how to prove the theorem. Thanks.

Answer 1

There is a theorem which follows from induction called the Well Ordering Principle, which says that any non-empty set of positive integers has a least element. It is easy to show that a least element is unique.

Given two non-zero integers, $m,n$, the set of positive common multiples is non-empty since it contains $|mn|$. So the least common multiple must exist and be unique.

Answer 2

The proof depends on your definition of lcm. If it is defined as the least element of the set of positive common multiples then uniqueness is clear, since if $\rm\:m,n\:$ are both lcms then by the leastness of both $\rm\: m\le n,\, n\le m\:\Rightarrow\: n = m,\,$ since $\rm\,n\,$ is sandwiched between $\,\rm m\,$ and $\,\rm m.\,$ For existence, the set of common multiples of a nonempty finite set omitting $\,0\,$ is always nonempty, since the product of the elements of the set is a nonzero common multiple of each element (nonzero since a product of nonzero elements remains nonzero in any integral domain).

If, instead, you use the universal definition $\rm\: a,b\mid c\iff lcm(a,b)\mid c\:$ then the uniqueness follows similarly, i.e. if $\rm\,m,n\,$ are lcms of $\rm \,a,b\,$ then choosing $\rm\,c=m\,$ and $\rm\,c=n\,$ above yields that $\rm\:m\mid n,\, n\mid m,\:$ so $\rm\:m = n\:$ if lcms are normalized to be positive. But in general domains we can only conclude the lcm is unique up to a unit (invertible) factor $\,\rm u,\,$ i.e. $\rm\,n = um,\,$ and there may be no effective algorithm to choose such normalize unit multiples, such as choosing positives in $\Bbb Z$ or monics (lead coef $=1$) in $\,\Bbb Q[x] =\,$ polynomials in $x$ with rational coef's.

The universal definition is preferred since not only does it work in any domain, but it helps to greatly simply diverse proofs involving gcds and lcms (e.g. here). It is a prototyical example of a point that I often emphasize: uniqueness theorems provide powerful tools for proving equalities.