Prime divisors of n!

Dan Bergren 29 Sep 2008 07:32

I was working on the homework for Tuesday and I noticed that the prime divisors of 15! are the same as the prime divisors of all integers between 1 and 15 (inclusive). These prime divisors are also the same as all primes between 1 and 15.

I think we can generalize for any n, the prime divisors of n! are the same as all primes p where 1$<$p$\le$n.

This is true because n, n-1, …, 3, 2 all possess prime divisors that are less than themselves, and thus must be less than the n = max{n, n-1, …, 3, 2}. Also, all p$\le$n are included because they are included as integers in the factorial of n!.