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Math Help - Solving congruences

  1. #1
    Junior Member Nerdfighter's Avatar
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    Washington
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    Solving congruences

    I understand that to solve ax congruent b modulo n means finding integers c and m such that ax congruent b modulo n is equivalent to x congruent c modulo n.

    But.... I don't know how to get there.
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  2. #2
    Super Member Gamma's Avatar
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    Dec 2008
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    Iowa City, IA
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    goal: solve ax \equiv b (mod n)
    Note this has solutions iff gcd(a,n)=d|b. This implies b=kd.

    step 1. Use the Euclidean Algorithm to find integers s and t that satisfy:
    as + nt = d

    step 2. look at this congruence mod n, and notice you get:
    as \equiv d (mod n)

    step 3. Multiply both sides by k to get:
    a(sk) \equiv dk=b (mod n)
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