Question

Question

Prove that n^2-n is divisible by 2 for every positive integer n

Answer 1

Suppose the positive integer is n.

∴ n = 2q or n = 2q + 1 where q∈Z.

CASE 1:-
n = 2q
∴ n² - n = (2q)² - 2q
= 4q² - 2q
= 2(2q² - q)

CASE 2:-
n = 2q + 1
∴n² - n = (2q + 1)² - (2q + 1)
= 4q² + 4q + 1 - 2q - 1
= 4q² + 2q
= 2(2q² + q)

Thus, in any case, n² - n is divisible by 2.

Thus, n² - n is divisible by 2 for every positive integer n.