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Question

Question 7
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

Let x = 2m + 1 and y = 2k + 3 are odd positive integers, for some positive integer m, k.
Then, x2+y2=(2m+1)2+(2k+3)2
= 4m2+1+4m+4k2+9+12k [ (a+b)2=a2+2ab+b2]
= 4(m2+k2)+4(m+3k)+10= even
= 4[(m2+k2)+(m+3k)+2]+2= even
On dividing x2+y2 by 4 , it leaves 2 as the remainder [division algorithm]
x2+y2 is not divisible by 4 .
Hence, x2+y2 is even for every odd positive integer x and y but not divisible by 4.

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