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

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Solution:
Since x and y are odd positive integers, we have
x = 2m + 1 and y = 2n + 1
=> x2 + y2 = (2m + 1)2 + (2n + 1)2
= 4m2 + 4m + 1 + 4n2 + 4n + 1
= 4(m2 + n2) + 4(m + n) + 2
Hence, x2 + y2 is an even number but not divisible by 4.

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