Proving Divisibility: The Relationship between n, p, and (n-1)!

[annoymage]

Homework Statement

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

Homework Equations

n/a

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

 

[Mark44]

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.

 

[annoymage]

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

 

[Mark44]

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.

 

[annoymage]

i'll try by induction, thank you ^^