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