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  • Prove that if x and y are both odd positive integers, then x square 2 + ysquare 2 is even but not divisible by 4.

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

Answers (1)

We can represent any positive integer in the form of (2q +1) for any integer q.
Let     x = 2q + 1; y = 2p +  1
x2 + y2 = (2q + 1)2 + (2p + 1)2
x2 + y2 = (2q)2 + (1)2 + 2(2q) (1) + (2p)2 + (1)2 + 2(2p) (1)
(Using (a + b) 2 = a2 + b + 2ab)
x2 + y2 = 4q2 + 1 + 4q + 4p2 + 1 + 4p
x2 + y2 = 2(2q2 + 2q + 2p2 + 2p + 1)
(2q2 + 2q + 2p2 + 2p + 1) is odd
Hence, x2 + y2 is divisible by 2 but not divisible by 4.

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infoexpert27

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