Cool property of the number $24$

Question

Recently I've had my 24th birthday, and a friend commented that it was a very boring number, going from 23 which is prime, 25 which is the first number that can be written as the sum of 2 different pairs of squared integers $3^2+4^2 =0^2+5^2 =25$, 24 seems like a very boring number

however, it seems to have a very special property

Theorem: product of 4 positive consecutive numbers is divisible by 24.

I managed to prove this via long and dry induction, not very interesting. I wonder if anyone can propose a different more elegant and witty proof, rather than dry algebra like me.

Answer 1

In$\;4\;$ consecutive integers $\;(n-1)\,,\,n\,,\,(n+1)\,,\,(n+2)\;$ there are exactly two even and two odd ones.

Of the even ones, exactly one is divisible by $\; 4\;$ so the whole product is divisible by $\;8\;$ , and since at least one of the four numbers is a multiple of three the whole thing is divisible by $\;2^3\cdot 3=24\;$ .

Answer 2

We have $$n(n-1)(n-2)(n-3)=\frac{n!}{(n-4)!}=4!\times\frac{n!}{4!(n-4)!}=24\times\binom{n}{4},$$ where the binomial coefficient $\binom{n}{4}$ is known to be an integer.

For more, see these previous questions: 1, 2, 3

Answer 3

24 is a very special integer number in many regards. John Baez has a nice pdf file about it: [ http://math.ucr.edu/home/baez/numbers/24.pdf ], and there is a very good Youtube video you can watch: https://www.youtube.com/watch?v=vzjbRhYjELo