# More on Wilson and Fermat

• Jul 23rd 2008, 09:16 AM
kel1487
More on Wilson and Fermat
1. If n E Z n>1 . N is prime if and only if (n-2)!= 1 modn

2. Let n be composite and >4. Prove (n-1)! = 0mod n

3. Show that 11 divides 456^654 +123^321

4. Find the solution to 9x=21mod23

Thanks for any help!
• Jul 23rd 2008, 09:42 AM
ThePerfectHacker
Quote:

Originally Posted by kel1487
1. If n E Z n>1 . N is prime if and only if (n-2)!= 1 modn

If n is prime then (n-1)!=-1 (n). Write as (n-1)(n-2)!=-1 (n). But n-1=-1 (n). Thus, we are left with after dividing (n-2)!=1 (n).

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2. Let n be composite and >4. Prove (n-1)! = 0mod n
See this.

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3. Show that 11 divides 456^654 +123^321
Find 456^654 (mod 11) and 123^321 (mod 11). Now so their remainders add up to a multiple of 11.

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4. Find the solution to 9x=21mod23
Note gcd(9,23)=1 and 1|21 so there is a unique solution.
Divide by 3 to get, 3x=7(mod 23) which is equvailent to 3x=30(mod 23).
Thus, we get x=10(mod 23).
• Jul 23rd 2008, 08:07 PM
kel1487
Thanks for all of your help!!