Question

# Prove that if x and y are odd positive integers, then x^2+y^2is even but not divisible by 4.

Solution

## 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 since 4 and 2 are multiples of 2 , x2 + y2 is a multiple of 2, x2 + y2is an even number 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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