Question

prive that if x and y are both odd positive integers then x^2+y^2 is even but not divisible by 4​

Answer 1

Let the two odd positive no. be x = 2k + 1 and y = 2p + 1

Hence, x2 + y2 = (2k + 1)2 +(2p + 1)2

= 4k2 + 4k + 1  + 4p2 + 4p + 1

= 4k2 + 4p2 + 4k + 4p + 2

= 4 (k2 + p2 + k + p) + 2  

The sum of square is even the no. is not divisible by 4

Hence, if x and y are odd positive integer, then x2 + y2 is even but not divisible by four.

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Answer 2

Let the two odd positive no. be x = 2k + 1 and y = 2p + 1

Hence, x2 + y2 = (2k + 1)2 +(2p + 1)2

= 4k2 + 4k + 1  + 4p2 + 4p + 1

= 4k2 + 4p2 + 4k + 4p + 2

= 4 (k2 + p2 + k + p) + 2  

The sum of square is even the no. is not divisible by 4

Hence, if x and y are odd positive integer, then x2 + y2 is even but not divisible by four.