Question

If \( x \) and \( y \) are any positive integers, then \( \left(x^{2}-x\right)+\left(y^{2}-y\right) \) is always. \( (x \neq 1, y \neq 1) \) 

(A) even number 

(B) odd number 

(C) prime number 

(D) both even and odd are possible

Answer 1

(x² - x) + (y² - y)

 = x(x - 1) + y(y - 1)

=  even

\(\because\) x(x - 1) is always even for an integer.

\(\therefore\) x(x - 1) is even & y(y - 1) is even

⇒ x(x - 1) + y(y - 1) is even

⇒ (x² - x) + (y² - y) is even