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

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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