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Math Help - More on Wilson and Fermat

  1. #1
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    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


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  2. #2
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    Quote Originally Posted by kel1487 View Post
    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).

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

    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.

    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).
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  3. #3
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    Thanks for all of your help!!
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