# Thread: Let p be odd. Then 2(p-3)!\equiv -1 (mod p)

1. ## Let p be odd. Then 2(p-3)!\equiv -1 (mod p)

Let $p$ be odd. Then $2(p-3)!\equiv -1 \ \mbox{(mod p)}$

Don't know how to do this one.

2. Originally Posted by dwsmith
Let $p$ be odd. Then $2(p-3)!\equiv -1 \ \mbox{(mod p)}$

Don't know how to do this one.
$2(p-3)!=(-1)(-2)(p-3)!\equiv(p-1)!\equiv-1\bmod{p}$

3. Originally Posted by chiph588@
$(-1)(-2)(p-3)!\equiv(p-1)!$
How do you show those two are congruent though?

4. Originally Posted by dwsmith
How do you show those two are congruent though?
$p-a\equiv-a\bmod{p}$

5. Originally Posted by chiph588@
$p-a\equiv-a\bmod{p}$
I don't see the connection.

6. Originally Posted by dwsmith
I don't see the connection.
What are you having trouble with?