<s>[INST] prove that the product of 4 consecutive positive integers is divisible by 24 How can I prove that prove that the product of $4$ consecutive positive integers is divisible by $24$, ie for any positive integer $n$ : $n(n+1)(n+2)(n+3)$
is divisible by $24$.
I've noticed that:
$24$ = $2^3 * 3$
$n(n+1)(n+2)(n+3)$ is divisible by $2*2$ so by $4$ (as there are at least 2 even numbers, obvious)
$n(n+1)(n+2)(n+3)$ is divisible by $3$ (as there is the product of 3 consecutive integers, easily provable using congruences)
so I've proved that you can divide it by $12$ until now. What am I missing here?
 [/INST] Notice that among $4$ consecutive numbers exactly one is divisible by 4 and one more divisible by 2 (say $n$ and $n+2$ or $n+1$ and $n+3$).
Notice also that among $3$ consecutive numbers exactly one is divisible by 3.
So the number is divisible by $3\cdot 8$.
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