Question

**Claim: **For any natural number n, if 6 ∣ n, then 3 ∣ n.

Step 1

Use the mathematical induction to show that the given statement **“****For any natural number n, if 6 ****∣**** n, then 3 ****∣**** n”** is true for all natural numbers **n. **

** **

Let the number **6 **can be written in the form **n(n+1)(n+2) **because **n(n+1) (n+2) **is always divisible by **6.**

** **

Using mathematical induction:

Step 2

let the statement is true for **n = k so k(k+1)(k+2) **is divisible by **3.**

** **

Then the statement **n(n+1) (n+2) **is always divisible by **6 **is true for **n = k+1.**

** **

**So,**

Step 3

**So, **the above statement is true as the above expression must be divisible by **6**. And we have to show that the same expression is divisible by 3. The expression **3(k2+3k+2)** is always divisible by **3** and the other expression **k(k2+3k+2) = k(k+1)(k+2)**

So,

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