Question

PROVE THAT (n²-n)IS DIVISIBLE BY 2 FOR EVERY POSITIVE INTEGER n

Answer 1

Casei: Let n be an even positive integer.
When n = 2q
In this case , we have
n² - n = (2q)² - 2q = 4q² - 2q = 2q (2q - 1 )
n² - n = 2r , where r = q (2q - 1)
n² - n is divisible by 2 .

Case ii: Let n be an odd positive integer.
When n = 2q + 1
In this case
n² -n = (2q + 1)² - (2q + 1)= (2q +1) ( 2q+1 -1)= 2q (2q + 1)
n² - n = 2r , where r = q (2q + 1)
n² - n is divisible by 2.

∴ n ² - n is divisible by 2 for every integer n