1)Suppose that . So and are divisible by . Then and are divisible by , that is is divisible by .
So if , then .
2)Suppose that is divisible by some prime . Then and are divisible by .
Since is divisible by , there is such that .
Since is divisible by , we have that is divisible by or is divisible by .
If is divisible by , then for some , then , that is is divisible by .
By symmetry we obtain that if is divisible by , then is divisible by .
In both cases and are divisible by , so is divisible by .
Therefore if , then .
From 1) and 2) we obtain that if and only if .
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