Question

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

Answer 1

To Prove: For any positive integer n, n³ − n is divisible by 6.

Proof: Let n be any positive integer.

Since any positive integer is of the form 6q or 6q + 1 or 6q + 2 or 6q + 3 or 6q + 4, 6q + 5

If n = 6q

If n = 6q + 1

If n = 6q + 2

Similarly we can prove others.

Hence it is proved that for any positive integer n, n³ − n is divisible by 6.