Question

The least common multiple of nonzero integers a and b is the smallest positive integer m...

The least common multiple of nonzero integers a and b is the smallest positive integer m such that a | m and b | m; m is usually denoted [a,b]. Prove that

[a,b] = ab/(a,b) if a > 0 and b > 0.

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Answer #1

Proof:

If  , we can write   and .

divides    . Similarly , divides .  

Hence   is a common multiple of   and   and  .

Let   be any positive common multiple  of   and   so that   and   for some  

Since    , there are integers   such that  .

Hence ,   divides

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