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Question

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

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Solution

Let x = 2m + 1 and y = 2m + 3 are odd positive integers, for every positive integer m,
Then, x2+y2=(2m+1)2+(2m+3)2
= 4m2+1+4m+4m2+9+12m[(a+b)2=a2+2ab+b2]
=8m2+16m+10=even
=2(4m2+8m+5)or4(2m2+4m+2)+1
Hence, x2+y2 is even for every positive integer m but not divisible by 4.

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