Prove that if x and y are both odd positive integers, then x2 + y2 i

1.	Prove that if x and y are both odd positive integers, then x2 + y2 is even but not divisible by 4.
Answer» Let the two odd positive numbers x and y be 2k + 1 and 2p + 1, respectively i.e., 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 Thus, the sum of square is even the number is not divisible by 4 Therefore, if x and y are odd positive integer, then x² + y² is even but not divisible by four. Hence Proved

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

Answer»

Let the two odd positive numbers x and y be 2k + 1 and 2p + 1, respectively

i.e., 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

Thus, the sum of square is even the number is not divisible by 4

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

Hence Proved