Question

Prove that nº-n is divisible by 2 for every positive integer n.​

Answer 1

Answer:

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.

Step-by-step explanation: