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Abstract Alg. Proof w/ Mod Congruence and Relative Primes

  1. Feb 22, 2015 #1
    1. The problem statement, all variables and given/known data
    If ≡(mod), ≡(mod),and gcd(,)=1,provethat ≡ (mod ).


    2. Relevant equations
    If ≡(mod)→n|ab-cd
    ≡(mod)→n|b-d
    gcd(,)=1→ relatively prime. So bx+ny=1

    Need to show n|a-c→a-c=nw

    3. The attempt at a solution
    If n|ab-cd, then nk=ab-cd
    If n|b-d, then nl=b-d
    If n|ab-cd AND n|b-d, then n|p(ab-cd)+q(b-d). So pab-pcd+qb-qd.
    pad-pcd+qb-qd
    =pab+qb-pcd-qd
    =b(pa+q) +d(-pc-q)

    I'm stuck! Don't know if this is going anywhere.
     
  2. jcsd
  3. Feb 23, 2015 #2

    Stephen Tashi

    User Avatar
    Science Advisor

    You need to edit your post so the variables show up. ( https://www.physicsforums.com/help/latexhelp/ )
    Do you mean something like [itex] ab \equiv cd \ (mod \ n) [/itex] ?
     
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