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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 x+ y2 = (2k + 1)2 + (2p + 1)2
                     = 4k2 + 4k + 1 + 4p2 + 4p + 1
                     = 4k2 + 4p+ 4k + 4p + 2
                     = 4(k2 + p+ k + p) + 2
since 4 and 2 are multiples of 2 , x+ y2 is a multiple of 2, x+ 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 x+ y2 is even but not divisible by 4
 

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