(n³ − n) is divisible by 3. Explain the reason.

Given, (n³ – n) is divisible by 3

Here, (n³ – n) can be written as n(n² -1)

⇒ n(n² -1²)

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

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

It is the product of three consecutive numbers

The product of three consecutive numbers is divisible by 3

Example:

Let n = 2

Substitute in the above equation we get

⇒ (2-1)(2)(2 + 1)

= 1 × 2 × 3

= 6 which is divisible by 3

Hence, (n³ – n) is divisible by 3