Show that if m and n are integers such that 4|m2+n2, then 4|mn
The Attempt at a Solution
Since 4 divides m2+n2, then we can say that m2+n2 = 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 m2+n2 = 4k. To do this, I multiplied and divided the left-hand side of the equation by mn to get: mn(m2+n2/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/(m2+n2/mn) = mn, or 4kmn/(m2+n2) = mn. However, I don't think this proves anything. Any advice on how to proceed (or even where to start)?