Question

Prove that if x and y are odd positive integers, then x square plus y square is even but not divisible by 4

Answer 1

Let the two odd positive numbers be x = 2k + 1 a nd y = 2p + 1

Hence x2 + y2 = (2k + 1)2 + (2p + 1)2

                    = 4k2 + 4k + 1 + 4p2 + 4p + 1

                    = 4k2 + 4p2 + 4k + 4p + 2

                    = 4(k2 + p2 + k + p) + 2

Clearly notice that the sum of square is even the number is not divisible by 4

Hence if x and y are odd positive integers, then x2 + y2 is even but not divisible by 4

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