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Modular arithmetic

  1. Jul 3, 2012 #1
    1. The problem statement, all variables and given/known data

    Hi everyone, I have a problem in the following modular arithmetic operation

    1/2*(x-4)(x-5) = 4(x-4)(x-5) (mod 7) ("=" means congruent in this expression)

    2. Relevant equations



    3. The attempt at a solution

    I am completely lost on how the operation is valid. If the coefficient is an integer between 0 to 6 then I can easily find its inverse using euclid extended algorithm but the problem here is that the coefficient is 1/2, a rational number. Any help on this problem will be greatly appreciated!
     
  2. jcsd
  3. Jul 3, 2012 #2
    4 x 2 = 1 (mod 7)

    So 1/2 = 4 (mod 7)

    The integers mod 7 are a field so you can always divide by any nonzero number.

    Another way to think of it is that 1/2 is 1 * 2-1. What is the multiplicative inverse of 2? It's 4. So 1/2 = 4 (mod 7).
     
  4. Jul 3, 2012 #3

    HallsofIvy

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    1/2= 4 (mod 7) because 4(2)= 8= 1 (mod 7) and that is, after all, the definition of "multiplicative inverse".
     
  5. Jul 3, 2012 #4
    Oh alright i think I'm starting to get it. So if the question is 1/3*(x-4)(x-5) it is congruent to 5(x-4)(x-5) (mod 7) because 3 * 5 = 1 mod 7?

    And what if the denominator is larger than the mod, say 1/30*(x-4)(x-5), should it be:
    30^-1 * (x-4)(x-5)
    and 30* 4 = 1 mod 7
    therefore 1/30*(x-4)(x-5) = 4 * (x-4)(x-5) mod 7 ?
     
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