Discrete Math - a modulus proof

  1. 1. The problem statement, all variables and given/known data

    I have to prove the following claim.

    Claim: For any positive integers m and n, m and n both greater than 1, if n|m and a≡b(mod m), then a≡b(mod n).


    2. Relevant equations

    n/a


    3. The attempt at a solution

    so i first changed each equation (ex: a≡b(mod m)) to a=b+qm and a=b+qn

    I figured in these forms I could show that the equations are equal.

    so I eventually get (a-b)/q=m or =n respectively. So I believe this shows their equality, but i am completely unsure because it won't always work I don't think. I need to also show that n|m. So tired dividing the m=(a-b)/q by the n= equation and of course I just get 1...

    To be completely honest I am not quite sure how to prove this. I am not quite familiar with the mod function and I am incredibly weak with proofs. If anyone can give me insight into solving this problem I would great appreciative.

    also note that this should be able to be done with a direct proof.

    thanks!
     
  2. jcsd
  3. Dick

    Dick 25,852
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    a=b(mod m) means m|(a-b). a=b(mod n) means n|(a-b). If n|m what can you conclude from this?
     
  4. I guess you could conclude that (m|(a-b)) / (n|(a-b)). Or I guess that's like m|n? Actually I'm not really sure. I'm not sure if that's correct. If it is, I'm not really sure if that proves the statement. If m|n is true, does that mean they're the same integer? In which case I assume that would prove it.
     
  5. Dick

    Dick 25,852
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    m|(a-b) and n|(a-b) are true/false statements. You can't DIVIDE them. Think about this. If a|b and b|c then a|c, right?
     
  6. Right, i mean that makes sense, but that's just reiterating the claim isn't it?

    If n|m and m|(a-b), then n|(a-b) is like saying If a|b and b|c then a|c.

    I think that there is some constant k so that kn=m? Then in this case, because they're modular, they come up with the same answer. Is that any more correct?
     
  7. Dick

    Dick 25,852
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    Yes. I just really didn't like (m|(a-b)) / (n|(a-b)). Didn't make much sense to me. Maybe it did to you.
     
    Last edited: Sep 16, 2008
  8. I see, but this is where my weak point is. I'm not sure how I write a proof for that.
     
  9. Dick

    Dick 25,852
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    "If n|m and m|(a-b), then n|(a-b) is like saying If a|b and b|c then a|c." That's the basis of the proof. Can you just flesh that out into a full proof? Start with "if n|m and a≡b(mod m)" and conclude with "then a≡b(mod n)".
     
  10. HallsofIvy

    HallsofIvy 40,534
    Staff Emeritus
    Science Advisor

    If a= b mod m, then m divides a- b so a-b= km for some integer k. If n divides m then m= pn for some integer p. Put those together to get that a-b= ?n.
     
  11. alright, thanks you two. Hopefully that'll help. I'll let you know if I have any further questions. Thanks! :)
     
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