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> Use induction to show that for any natural number $n\ge 1$, given pairs $(a\_1,b\_1),(a\_2,b\_2),\ldots,(a\_n,b\_n)$ of integer numbers, there exist integer numbers $c$ and $d$ such that $(a\_1^2+b\_1^2)(a\_2^2+b\_2^2)\cdots(a\_n^2+b\_n^2)=c^2+d^2$.
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At first I tried expressing it like the Pythagorean Theorem, but that doesn't work for all squares. I checked out the base case, and I'm on the inductive step. I can't seem to figure it out. I'm only one month into my Discrete class, so the answer shouldn't be anything terribly complicated. I would appreciate any and all help.