Question

prove that if x and y are odd positive integers;x>y then x^2-y^2 divisible by 4​

Answer 1

Answer:

Since x is odd, then x = 2a + 1 for some integer a.

Since y is odd, then y = 2b + 1 for some integer b.

Notice x>y means a>b, but that is actually not important here!

In terms of a and b, we have

   x² - y²

= ( 2a + 1 )² - ( 2b + 1 )²

= ( 4a² + 4a + 1 ) - ( 4b² + 4b + 1 )

= 4 ( a² + a + b² + b )

which is divisible by 4 since (a²+a+b²+b) is an integer.

Hope this helps!