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Question about simplifying congruences

  1. Nov 14, 2012 #1
    1. The problem statement, all variables and given/known data
    I am looking at an example from my book and I have no clue how this is done.


    2. Relevant equations
    I solved the left hand side and got this
    125t3+75t2+50t+4

    How did they reduce it to 65t+5?


    3. The attempt at a solution
    http://i48.tinypic.com/2v2gpap.jpg
     
  2. jcsd
  3. Nov 15, 2012 #2

    SammyS

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    [itex]125\equiv 0\ (\!\!\!\!\mod25) [/itex]

    etc.
     
  4. Nov 15, 2012 #3
    I don't understand please clarify
     
  5. Nov 15, 2012 #4

    micromass

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    Do you understand what (mod 25) means?
     
  6. Nov 15, 2012 #5
    I don't understand where he got the 125 and what happened to the t
     
  7. Nov 15, 2012 #6

    micromass

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    But you're asking why

    [tex]125t^3+75t^2+50t+4 = 65t+5 ~\text{(mod 25)}[/tex]

    right??

    So, I ask: do you know what (mod 25) means?
     
  8. Nov 15, 2012 #7

    SammyS

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    If [itex]\ 125\equiv 0\ (\!\!\!\!\mod25)\ [/itex], then [itex]\ 125t^3\equiv 0t^3\ (\!\!\!\!\mod25)[/itex]

    You do understand that [itex]\ 125\equiv 0\ (\!\!\!\!\mod25)\,,\ [/itex] don't you?
     
  9. Nov 15, 2012 #8
    That means 25|125, sorry I guess the variables just confuses me.
     
  10. Nov 15, 2012 #9

    SammyS

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    Anyway ...

    Do you now understand why [itex]\displaystyle \ \ 250 t^3+150 t^2+65 t+ 5 \equiv 65t+5 ~\text{(mod 25)}\ ?[/itex]

    Yes, I get [itex]\displaystyle \ \ 2(1+5t)^3+7(1+5t)-4= 250 t^3+150 t^2+65 t+ 5 \ .[/itex] ​
     
  11. Nov 15, 2012 #10
    Is it because 25|250 and 25|150 so the remainder is 65t+5?
     
  12. Nov 15, 2012 #11

    SammyS

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    Well, the remainder of 65t+5 is the same as the remainder of 250t3+150t2+65t+5 when divided by 25 .
     
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