Prove that n^3-n is divisible by 3

Given

n³ – n is divisible by 3 ∀ n ∈ N

Proof

We will prove the given statement by induction

STEP 1

n = 1

n³ – n = 1³ – 1 = 0

0 is divisible by 3. Therefore, the statement is true for n = 1

STEP 2

Let the given statement be true for n = k

k³ – k = 3x

Now, we need to prove that if the statement is true for n = k then it is also true for n = k + 1

(k + 1)³ – (k + 1) = k³ + 1 + 3k² + 3k – k – 1

= k³ – k + 3k² + 3k

= 3x + 3(k² + k)

= 3 * ( x + k² + k )

We can say that (k + 1)³ – (k + 1) is divisible by 3 if k³ – k is divisible by 3.

Therefore, we can say that if the given statement is true for n = k, then it is also true for n = k + 1. Hence, by the principle of mathematical induction, the given statement is true ∀ n ∈ N.