Question

If $x$ and $y$ are positive integers, and it $x-y$ is even, show that $x^2-y^2$ is divisible by $4$.

Answer 1

Given $x-y$ is even ;

Every number is of the form $2k$ or $2k+1$. For difference of two numbers to be even both should be either $2k$ form or $2k+1$ form.

ie. both $x$ and $y$ are even or both $x,y$ are odd.

Both $x,y$ are even say $x=2k_1;y=2k_2$

$x^2-y^2=4k_1^2-4k_2^2=4(k_1^2-k_2^2)$ which is divisible by 4.

Both $x,y$ are odd say $x=2k_1+1;y=2k_2+1$

$x^2-y^2=4k_1^2+4k_1+1-4k_2^2-4k_2-1=4(k_1^2+k_1-k_2^2-k_2)$ which is divisible by $4$.

Thus if $x-y$ is even $x^2-y^2$ is divisible by $4$