If 5 divides a^2+b^2+c^2, then 5 divides a or 5 divides b or 5 divides c.
I figured I would use the contrapositive. So I suppose 5 does not divide a nor b nor c. So each of a, b, and c are of the form 5k+1 or 5k+2 or 5k+3 or 5k+4. One method would involve taking all possibilities for each of a,b,c and showing that 5 does not divide the result. The problem is that I get something like 4^3 cases to compute, and so, this would take forever. Is there a simpler way?