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 $(a_1^2+b_1^2)(a_2^2+b_2^2)\cdots(a_n^2+b_n^2)=c^2+d^2$.
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.