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.

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