- #1

- 185

- 4

## Homework Statement

Show that if m and n are integers such that 4|m

^{2}+n

^{2}, then 4|mn

## Homework Equations

## The Attempt at a Solution

Since 4 divides m

^{2}+n

^{2}, then we can say that m

^{2}+n

^{2}= 4k, where k is an integer. I haven't done any mathematical proofs of any kind yet, but we were supposed to see if we could do this. I am kind of stuck as to where to go from here. The only thing I could think of is to try to introduce a m*n term into the equation m

^{2}+n

^{2}= 4k. To do this, I multiplied and divided the left-hand side of the equation by mn to get: mn(m

^{2}+n

^{2}/mn) = 4k. Since I am looking to show that 4|mn, or mn = 4*(some integer), I tried to isolate the mn term. That means 4k/(m

^{2}+n

^{2}/mn) = mn, or 4kmn/(m

^{2}+n

^{2}) = mn. However, I don't think this proves anything. Any advice on how to proceed (or even where to start)?

Thanks.