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Math Help - more modulo proofs

  1. #1
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    more modulo proofs

    any help on getting started with these would be great. I seem to be having a dead brain day.


    • Let p be a prime number.Suppose you take all the non zero integers 0,1,2,...,p-1, multiply them together, and take the answer modulo p.Show that you will always get p-1



    • Determine which of 1,2,3,4,5,6,7,8,9,10 (the non zero integers modulo 11) are squares modulo 11. Use this to find the roots of x^5 - 1 modulo 11



    • Let f(x) be a polynomial with integer coefficients. Show that if the value of f(x) is divisible by r at r consecutive integers, then f(m) is divisible by are for all integers m.


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  2. #2
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    Quote Originally Posted by gpenguin View Post
    any help on getting started with these would be great. I seem to be having a dead brain day.


    • Let p be a prime number.Suppose you take all the non zero integers 0,1,2,...,p-1, multiply them together, and take the answer modulo p.Show that you will always get p-1



    • Determine which of 1,2,3,4,5,6,7,8,9,10 (the non zero integers modulo 11) are squares modulo 11. Use this to find the roots of x^5 - 1 modulo 11



    • Let f(x) be a polynomial with integer coefficients. Show that if the value of f(x) is divisible by r at r consecutive integers, then f(m) is divisible by are for all integers m.


    Thank you

    What've you done so far?

    Tonio
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  3. #3
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    • let p be a prime number...


    for p= 3
    2 (mod 3) = 1

    which isn't true and then confuses me



    • determine which of ...

    I don't know what squares modulo 11 means



    • let f(x)=...


    eg r= 2
    if f(m) and f(m+1) are divisible by 2 then f(m) is always divisible by 2

    and then i don't know how to compute this
    I need to do r=2 and r=3
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  4. #4
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    2(mod 3) = 2.....not 1.
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  5. #5
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    That helps with proving through numbers. But I still can't prove it generally.
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  6. #6
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    Quote Originally Posted by gpenguin View Post
    • Determine which of 1,2,3,4,5,6,7,8,9,10 (the non zero integers modulo 11) are squares modulo 11. Use this to find the roots of x^5 - 1 modulo 11
    For example, 1*1 = 1 is a perfect square. 2*2 = 4 is a perfect square. Something a little less obvoius this time... 5*5 = 3, so 3 is a perfect square. Just run the list from 1*1 to 10*10 (mod 11). That will generate your list.

    -Dan
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