Prove that 2/n4-3 if and only off 4/n2+3
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Assume that This means that there is some integer such that Then
Since is an integer, we see that is odd.
Let Then is even if and only if is even.
Taking the contrapositive of both directions of the biconditional gives: is odd if and only if is odd.
Let Then is odd if and only if is odd. Then is odd if and only if is odd.
Therefore there is an integer such that
Since is an integer, we see that
Assume that This means that for either
(i) (ii) or (iii)
If is even then there is an integer such that
which implies that we must be in case (iii).
If is odd then there is an integer such that
In this case we see that This is contrary to our assumption that so under our assumption this case can never occur.
Thus we only have to consider case (iii)
Rearranging we can write Hence
This means that the remainder is when is divided by Therefore,
Therefore for if and only if