Question

Show that n(n-1) is divisible by 2 for every positive integer n

Answer 1

Case 1: 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 2: 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