Number theory, proving or finding counterexample.

Question

Prove or find a counterexample: The product of any three consecutive natural numbers is divisible by 6.

Answer

True, because product of three consecutive natural numbers can be divisible by 6. Thus, $(n)\dot{}(n+1)\dot{}(n+2) | 6$

Can i get a feedback for this answer.

Just use induction. – The very fluffy Panda Apr 10 '14 at 11:47 — The very fluffy Panda, Apr 10 '14 at 11:47

score 2 · Accepted Answer · answered Apr 10 '14 at 11:33

2

At least one of the $3$ consecutive numbers must be divisible by $2$. At least one (in fact, exactly one) of them must also be divisible by $3$.

If you want to be rigorous, write down the three natural numbers as $3n, 3n+1, 3n+2$ and show that at least one of them must be $0$ modulo $2$, and at least one of them must be $0$ modulo $3$.

Since both $2$ and $3$ divides the product, $2\cdot3=6$ must divide it as well.

answered Apr 10 '14 at 11:33

Yiyuan Lee

14,435

And this is a real proof, instead of restating the question – Mathias711 Apr 10 '14 at 11:34

Number theory, proving or finding counterexample.

1 Answers1