Prachi

If m and n are odd positive integers, then m ^ 2 + n ^ 2 is even, but not divisible by 4. Justify.

by

Let m be 2x + 1 and n be 2y + 1 
m ^ 2 + n ^ 2 = (2x + 1) ^ 2 + (2y + 1) ^ 2
                        = (2x) ^ 2 + (1) ^ 2 + 2(2x)(1) + (2y) ^ 2 + (1) ^ 2 + 2(2y)(1)
                        = 4x ^ 2 + 1 + 4x + 4y ^ 2 + 1 + 4y
                        = 4x ^ 2 + 4y ^ 2 + 4x + 4y + 2
                        = 2(2x ^ 2 + 2y ^ 2 + 2x + 2y + 1) , which is divisible by 2

Thus, m ^ 2 + n ^ 2 is an even integer.

Also,
m ^ 2 + n ^ 2 = 4x ^ 2 + 4y ^ 2 + 4x + 4y + 2
                        = 4(x ^ 2 + y ^ 2 + x + y) + 2
                        = 4q + 2 , where q = x ^ 2 + y ^ 2 + x + y

Thus, m ^ 2 + n ^ 2 leaves remainder 2 when divided by 4 and, hence, is not divisible by 4.