# Divisibility math problem

1. Oct 13, 2010

### annoymage

1. The problem statement, all variables and given/known data

i want to show if n = pq, 1 < p < n, then p l (n-1)!

2. Relevant equations

n/a

3. The attempt at a solution

i can see its true, because p < n, p l p, then p l (n-1)!. and this prove very ambiguous for me

2 question.

1.help me, i think there must be easier way to prove the question

2. btw, how do i prove those bold statement?

if 1 < p < n, p l p, then p l (n-1)! its something like this p l (1)....(p)....(n-1)!, help T_T

2. Oct 13, 2010

### Staff: Mentor

Re: divisibility

Here's an example (not a proof!) to show what's going on.

Let n = 21 = 7 * 3, with p = 7 and q = 3.

Does p | 20! ? Since 20! = 1 * 2 * 3 * ... * 6 * 7 * 8 * ... * 19 * 20, p clearly divides (n - 1)! in this example.

3. Oct 13, 2010

### annoymage

Re: divisibility

i know, that's what i already see. but how should i prove it T_T,

4. Oct 13, 2010

### Staff: Mentor

Re: divisibility

Show that, since p < n, then there is a factor of p in (n - 1)!. A proof by induction is one way to go. There might be a simpler way, but it doesn't occur to me.

5. Oct 13, 2010

### annoymage

Re: divisibility

i'll try by induction, thank you ^^