why the product of any four consecutive integers is divisible by 24

Question

anonymous · Answer

good question

directrix · Answer

From our friends at Wiki: "The product of any four consecutive numbers is divisible by 24. This is because among any four consecutive numbers there must be two even numbers, one of which is a multiple of four, and there must be a multiple of three." Note: Wikipedia is not considered to be an academic research site. http://en.wikipedia.org/wiki/24_(number)

across · Answer

Let $a_1$, $a_2$, $a_3$, and $a_4$ be any four consecutive, positive integers. Then we have $a_1=a_1$, $a_2=a_1+1$, $a_3=a_1+2$, and $a_4=a_1+3$. Multiplying them, we obtain$$a_1a_2a_3a_4=a_1(a_1+1)(a_1+2)(a_1+3)=$$$$a_1^4+6a_1^3+11a_1^2+6a_1.$$From here, you can begin a proof by induction: Notice that if we take $a_1=1$, the smallest positive integer, we will have $a_1a_2a_3a_4=24$, which is trivially divisible by $24$.

Can you finish the proof? Or were you just curious about this fact? :)