Prove that the invariant factors of $\mathbb Z_m\times\mathbb Z_n$ are $mn$ if $m$ and $n$ are relatively prime and are the greatest common divisor and the least common multiple of $m$ and $n$ if they are not relatively prime.
Invariant factors of $\mathbb Z_m\times\mathbb Z_n$
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abstract-algebra
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1Hint: use the Chinese Remainder Theorem. – 2012-11-22