Wilson theorem Question Explanation

[mathsss2]

How do I explain this:

Let p be odd prime explain why: 2*4*...*(p-1)\equiv (2-p)(4-p)*...*(p-1-p)\equiv(-1)^{\frac{(p-1)}{2}}*1*3*...*(p-2) mod p.



Relevant equations

Gauss lemma
wilson's theorem [(p-1)!\equiv-1 modp]



The attempt at a solution
pairing? need assistance


Thanks

 

[gabbagabbahey]

Hint: What are 2-(2-p), 4-(4-p), ... (p-1)-(p-1-p)? Are they divisible by p?

 

[mathsss2]

Thanks, this problem is solved.