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Math Help - Modular Proofs

  1. #1
    Newbie
    Joined
    Mar 2009
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    Modular Proofs

    I just can't seem to grasp modular arithmatic and I am trying to solve a few simple proofs:

    * means "is congruent"

    1) Prove if [m * n (mod 7)] then [m * N (mod 14)]

    2) Prove if [3 | n] then [n^(2) * 1 (mod 3)]

    Also does anyone know any good sites with some basic proofs like these with solutions? Any help is appreciated.
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  2. #2
    Senior Member
    Joined
    Apr 2009
    From
    Atlanta, GA
    Posts
    409

    Modular Arithmetic

    A few basic rules for ya:

    (i) Q \equiv R (\bmod D) literally means " Q leaves a remainder R when divided by D"
    (ii) Therefore, Q= kD+R for some integer k
    (iii) Alternately, D|(Q-R)
    (iv) The nice thing about modular arithmetic is that it follows pretty much all the same rules as regular arithmetic. For example, if a \equiv b (\bmod p) then 3a \equiv 3b (\bmod p), etc. So in most simple cases, you can treat congruences ( \equiv) like regular equations.

    1) Prove if [m * n (mod 7)] then [m * N (mod 14)]
    What is N? Does n=N?

    2) Prove if [3 | n] then [n^(2) * 1 (mod 3)]
    This statement is false. Counterexample: For n=6, 3|n. But n^2=36 \equiv 0 (\bmod 3)
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