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I am stuck on this problem, and was wondering if anyone could help me out with this. The question is as follows:

Let $n$ be an integer such that $n ≥ 1$. Prove that $6$ divides $n(n + 1)(n + 2)$.

Note: An integer $a$ divides an integer $b$, written $a|b$, if there exists $q ∈ Z$ such that $b = qa$. Alternatively, $a|b$ if dividing $b$ by $a$, $b ÷ a$, results in an integer.

Should I do a proof by induction?

All help/input is appreciated!

azimut
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choloboy
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    http://math.stackexchange.com/questions/12065/the-product-of-n-consecutive-integers-is-divisible-by-n-factorial or http://math.stackexchange.com/questions/12067/the-product-of-n-consecutive-integers-is-divisible-by-n-without-using-the-prop – lab bhattacharjee Oct 15 '13 at 16:10
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    Another way: http://math.stackexchange.com/questions/11601/proof-that-a-combination-is-an-integer – lab bhattacharjee Oct 15 '13 at 16:25
  • This question is also related. (If we denote $k=n+1$, then we want to show that $6$ divides $k(k-1)(k+1)=k(k^2-1)=k^3-k$.) – Martin Sleziak Dec 04 '15 at 12:55

7 Answers7

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$n$ is either even or odd. Also, it must be congruent, modulo $3$, to either $0$, $1$, or $2$.

dfeuer
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Another possibility:

$$\frac{n(n+1)(n+2)}{6}$$ equals the binomial coefficient $$\binom{n+2}{3}$$ which is an integer (since it is the number of 3-element subsets of an ($n+2$)-element set).

azimut
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The links provided by lab bhattacharjee solve the problem in more generality, but for the sake of completeness...

Let us do an inductive proof. For $n = 0$ things are trivial. Suppose we know the claim for $n$, and we want to prove it for $n+1$. By inductive assumption, $$n(n+1)(n+2) = n^3+2n^2 + 2n = 6 q$$ for some $q$. Now, $$(n+1)(n+2)(n+3) = n(n+1)(n+2) + 3(n+1)(n+2) = 6q + 3(n+1)(n+2)$$ A simple argument (use induction again, if you like) shows that $(n+1)(n+2)$ is divisible by $2$, so you can write $(n+1)(n+2) = 2p$. Consequently, $$(n+1)(n+2)(n+3) = 6(q+p)$$ which finishes the problem.

eem
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Jakub Konieczny
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    I think this answer could use an explanation of why induction is not a good approach to this problem. – dfeuer Oct 15 '13 at 20:27
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    I think any approach is good, as long as it works. Personally, I believe it is worth it to try and squeeze as much as possible of methods one is familiar with. If the person posing a problem thinks of induction - why not try it? It generalises quite nicely: Suppose you are trying to prove that $f(n)$ is divisible by $M$ for all $n$. Then just check if $M$ divides $f(0)$, and that $g(n) := f(n+1)-f(n)$ is divisible by $M$ for all $n$. For $g(n)$, if things are not already obvious, just apply the same reasoning. For instance, this proves little Fermat. – Jakub Konieczny Oct 16 '13 at 09:53
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    isn't it n(n + 1)(n+2) instead of (n + 2)(n + 3)n ? – RiaD Dec 18 '15 at 09:00
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The product $n(n+1)(n+2) \equiv 0\ (\mathrm{mod}\ 2)$, since if $n$ is even the product will be even, and if $n$ is odd then $n+1$ will be even and again the product is even.

We also know that $n(n+1)(n+2) \equiv 0\ (\mathrm{mod}\ 3)$. You can prove this by notating that $n\ \mathrm{mod}\ 3$ must be $0, 1,$ or $2$, and all of these cases imply the product is divisible by $3$. $$n\equiv0\ (\mathrm{mod}\ 3)\implies n(n+1)(n+2)\equiv0\ (\mathrm{mod}\ 3),$$ $$n \equiv1\ (\mathrm{mod}\ 3)\implies n+2\equiv 0\ (\mathrm{mod}\ 3)\implies n(n+1)(n+2)\equiv0\ (\mathrm{mod}\ 3),$$ $$n\equiv2\ (\mathrm{mod}\ 3)\implies n+1 \equiv 0\ (\mathrm{mod}\ 3) \implies n(n+1)(n+2)\equiv0\ (\mathrm{mod}\ 3).$$

Therefore $n(n+1)(n+2)$ is divisible by $2$ and by $3$, which implies it is divisible by $6$.

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Either one of $n$, $n+1$ and $n+2$ is divisible by $3$, because $n$ is either in the form of $3k$, $3k+1$ or $3k+2$. We also have either one of $n$ or $n+1$ is divisible by $2$ because they are two consecutive natural numbers. Thus, the product of $n$, $n+1$ and $n+2$ is always divisible by $2*3=6$.

CODE
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If a number is divisible by $6=2\cdot 3$, then it must be divisible by $2$, and also by $3$.

Look at $$n(n+1)(n+2)$$ if $n$ is not divisible by $2$, will any of the other factors be? If $n$ is not divisible by $3$ will any of the factors be?

cansomeonehelpmeout
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If we let $m=n+1$, we need to prove that $6 \vert m(m^2-1)$ for all $m \geq 2$. Equivalently, we want to show that $$m^3 \equiv m \bmod6$$ First note that $m \equiv 0,1,2,3,4,5 \pmod 6$. Now observe that following: $$m \equiv 0 \bmod 6 \implies m^3 \equiv 0^3 \bmod 6 \implies m^3 \equiv 0 \bmod6 $$ $$m \equiv 1 \bmod 6 \implies m^3 \equiv 1^3 \bmod 6 \implies m^3 \equiv 1 \bmod6$$ $$m \equiv 2 \bmod 6 \implies m^3 \equiv 2^3 \bmod 6 \implies m^3 \equiv 8 \bmod6 \implies m^3 \equiv 2 \bmod6$$ $$m \equiv 3 \bmod 6 \implies m^3 \equiv 3^3 \bmod 6 \implies m^3 \equiv 27 \bmod6 \implies m^3 \equiv 3 \bmod6$$ $$m \equiv 4 \bmod 6 \implies m^3 \equiv 4^3 \bmod 6 \implies m^3 \equiv 64 \bmod6 \implies m^3 \equiv 4 \bmod6$$ $$m \equiv 5 \bmod 6 \implies m^3 \equiv 5^3 \bmod 6 \implies m^3 \equiv 125 \bmod6 \implies m^3 \equiv 5 \bmod6$$ Hence, in all cases, we have $m^3 \equiv m \bmod 6$, which means that $$6 \vert (m^3-m) \implies 6 \vert n(n+1)(n+2)$$