Question

prove that if a and b are positive integers then a square+ b square is even but not divisible by 4

Answer 1

Let the two odd positive numbers be a = 2k + 1 a nd b= 2p + 1
Hence a2 + b2 = (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 a2 + b2 is even but not divisible by 4..

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