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Help me with algebra proof

  1. Sep 28, 2011 #1
    1. The problem statement, all variables and given/known data

    Question 1.

    http://people.math.carleton.ca/~mezo/A2math1102-11.pdf

    2. Relevant equations

    r1 = a modn
    r2 = b modn
    r = (a+b) modn



    3. The attempt at a solution


    I used the division algorithm

    So:

    a = (q1)n + r1
    b = (q2)n + r2
    (a+b) = (q3)n + r

    i isolated for r1 and r2 to get more equations

    r1 = a - (q1)n
    r2 = b - (q2)n
    r = (a+b) - (q3)n

    r1+r2 = (a+b) -n(q1+q2)

    I then added a and b together to get the following:

    a+b = r1+r2 + n(q1+q2)

    After that, i`m completely lost.

    This is for question 1 by the way. I've been toying around with this question for 15 hours and i still can't do anything with it. I thought i proved it before but then i realized it wasn't a proof. Please help. I'm dying.



    Also another thing that i'm having trouble with is 3. I don't understand why it's worth 10 marks. All i see is me setting equation 1 and 2 equal to each other. I don't understand what's so special about that.
     
    Last edited: Sep 28, 2011
  2. jcsd
  3. Sep 28, 2011 #2

    micromass

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    Substitute a and b from the first and second equation in the third equation. What do you get??



     
  4. Sep 28, 2011 #3
    (q1)n+(q2)n + r1 + r2 = (q3)n+r

    Is this what you meant?
     
  5. Sep 28, 2011 #4

    micromass

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    Yes, that is good. Now rearrange a bit and you got the proof!!
     
  6. Sep 28, 2011 #5
    I don't think it's that obvious to me. To be honest, i've already tried doing that and i'm not sure but maybe i don't truly know how mods truly work. if i isolated for r1 + r2, then i'd get the following:

    r1 + r2 = n(q3 - q2 - q1) + r

    r1 = amodn
    r2 = bmodn
    r = (a+b)modn

    I'm not exactly sure how it works i guess. Could you help me? It'd be greatly appreciated.
     
  7. Sep 28, 2011 #6

    micromass

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    Well, a=b (mod n) if there is a q such that qn=a-b. Agreed??

    Well, you've shown now that r1 + r2 - r = n(q3 - q2 - q1). Doesn't this prove it?
     
  8. Sep 28, 2011 #7
    To be honest, i don't really understand why a=b (mod n) if there is a q such that qn = a-b"

    Where'd that come from? Could you explain please? I'm sorry but it's really just not that obvious to me.
     
  9. Sep 28, 2011 #8

    micromass

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    What is the definition of a=b (mod n) ????
     
  10. Sep 28, 2011 #9
    a = q1n + a

    You mean that?
     
  11. Sep 28, 2011 #10

    micromass

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    Don't you mean [itex]a=q_1 n+b[/itex]??

    Well, that's the same thing I posted, no?? Just rearrange, and you get [itex]a-b=qn[/itex]...
     
  12. Sep 28, 2011 #11
    Actually i don't know but i think i posted it wrong. shouldn't it be ,

    b = qn + a, a = bmodn. That's how our professor wrote it in the form as atleast. Now i'm confused.


    I also really don't see how r1 + r2 - r = n(q3 - q2 - q1) proves it. Like I just don't see what this result means in relation to [itex]a-b=qn[/itex]. Like it's telling us to prove (a+b)modn = (amodn+bmodn)modn. I'm just not seeing the connection
     
    Last edited: Sep 29, 2011
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