Question

For any positive integer n, prove that n3 – n is divisible by 6.
​

Answer 1

Let :

a = n³ - n

= n(n² - 1)

= n(n - 1) (n + 1)

= (n - 1)n (n + 1)

1. Now, out of three (n - 1),n and (n + 1) one must be even so a is divisible by 2.
2. Also, (n - 1),n and (n + 1) are three consecutive integers thus as proved a must be divisible by 3.

From (1) and (2),

a must be divisible by 2 × 3 = 6

Hence, n³ - n is divisible by 6 for any positive integer n.