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Math Help - modulo ?

  1. #1
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    modulo ?

    hi there I having some troble with a few problems i was wondering if i could get some help?

    x^2 equvilent to -1 mod 19

    x^2 equvilent to -1 mod 19

    4x equvilent to 6 mod 15

    thanks for your help
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by bugal402 View Post
    4x equvilent to 6 mod 15

    thanks for your help
    I'll defer the first too to more able number theorists.

    4x\equiv 6\text{ mod }15\implies 4x=6-15y\implies 4x+15y=6 this is solvable since 6=6(15,4). Now do you know how to solve Diophantine equations?


    P.S. It might be easier to just guess :S
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  3. #3
    MHF Contributor chiph588@'s Avatar
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     (4,15)=1 \implies 4^{-1} exists modulo  15 .

    You can find  4^{-1} by the Euclidean algorithm.

    I omit details but  4\cdot4\equiv1\bmod{15}\implies 4^{-1}\equiv4\bmod{15} .

    So  x\equiv 4^{-1}\cdot4x\equiv 4^{-1}\cdot6\equiv 4\cdot6 = 24\equiv 9 \bmod{15}
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  4. #4
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    Quote Originally Posted by bugal402 View Post
    hi there I having some troble with a few problems i was wondering if i could get some help?

    x^2 equvilent to -1 mod 19

    x^2 equvilent to -1 mod 19


    You asked twice the same......anyway, the equation x^2=-1\!\!\!\pmod p has a solution iff p=1\!\!\!\pmod 4 , so in your case...

    Tonio


    4x equvilent to 6 mod 15

    thanks for your help
    .
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  5. #5
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    Quote Originally Posted by Drexel28 View Post
    I'll defer the first too to more able number theorists.

    4x\equiv 6\text{ mod }15\implies 4x=6-15y\implies 4x+15y=6 this is solvable since 6=6(15,4). Now do you know how to solve Diophantine equations?


    P.S. It might be easier to just guess :S
    I know how to do Diophantine equations. thanks again
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  6. #6
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    Quote Originally Posted by tonio View Post
    .
    you are right I did the second one was supposed to be x^2 equvilent to -1 mod 17

    sorry about that
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  7. #7
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    Quote Originally Posted by tonio View Post
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    So this one does not have a solution right?
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    Quote Originally Posted by bugal402 View Post
    So this one does not have a solution right?

    The one for 19 hasn't , and the one for 17 has...and a pretty easy one, in fact.

    Tonio
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