Will rate life saver if answered completly Thank YOU!!!! Let p be a prime, Then

Question

Will rate life saver if answered completly Thank YOU!!!!

Let p be a prime, Then 1 - 2,(p - 1) - 1(mod p) Suggestion: Try pairing numbers on the list 1,,p - 1 with their inverses mod p. What happens when you multiply (that is, multiply mod p) a number on the list with its inverse mod p? Note that for some numbers on the list, the inverse mod p is the same as the number. Which numbers on the list 1, 2,,p - 1 have this property? Using this information, find a way to reorganize the product 1, 2(p - 1).

Explanation / Answer

if p is prime, then each of the integers 1, 2, 3, 4, … , p1 are relatively prime to p.

It is easy to check the result when p is 2 or 3, so let us assume p > 3.

So for each of these integers a there is another b such that ab 1 (mod p).

It is important to note that this b is unique modulo p, and that since p is prime, a b if and only if a is 1 or p1.

Now if we omit 1 and p 1, then the others can be grouped into pairs whose product is 1 showing 2.3.4.….(p2) 1 (mod p).

Therefore we have:

1.2.3.4.….(p2).(p-1) 1.1.(p-1) (mod p)

and since (p-1) -1 (mod p)

We have:

1.2.3.4.….(p2).(p-1) -1 (mod p)

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