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

Question

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

Philoid · Accepted Answer

Let us consider two consecutive odd positive integers 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

Squaring these terms using (a + b)² = a² + 2ab + b²

⇒ x² + y² = 4m² + 1 + 4m + 4m² + 9 + 12m

⇒ x² + y² = 8m² + 16m + 10 , which is even

⇒ x² + y² = 2(4m² + 8m + 5)

⇒ x² + y² = 4(2m² + 4m + 2) + 2

Hence x² + y²is even for every positive integer m but not divisible by 4.

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