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Math Help - Chinese Remainder Theorem

  1. #1
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    Chinese Remainder Theorem

    have a problem on hand - this one is kinda easy- but am stuck :P

    the problem is :

    Solve using CRT:

    x congruent to 8(mod13)
    x congruent to 1(mod 5)
    x congruent to 5 (mod 6)

    i have gotten most of it. just want to be sure


    thanks
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  2. #2
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    Quote Originally Posted by darren_a1
    have a problem on hand - this one is kinda easy- but am stuck :P

    the problem is :

    Solve using CRT:

    x congruent to 8(mod13)
    x congruent to 1(mod 5)
    x congruent to 5 (mod 6)

    i have gotten most of it. just want to be sure


    thanks
    We have,
    x\equiv 8 \mod 13
    x\equiv 1 \mod 5
    x\equiv 5\mod 6
    Also, \gcd(13,5)=\gcd(5,6)=\gcd(13,6)=1
    thus, we may rely on Chinese Remainder Theorem.
    ----
    By Chinese Remainder Theorem, we know that,
    x\equiv a_1b_1N_1+a_2b_2N_2+a_2b_3N_3 \mod N
    Where,
    N=5\cdot 6\cdot 13
    N_1=N/13=30
    N_2=N/5=78
    N_3=N/6=65
    Also,
    a_1=8
    a_2=1
    a_3=5
    Finally,
    b_1\cdot N_1\equiv 1 \mod 13 thus, b_1=10
    b_2\cdot N_2\equiv 1\mod 5 thus, b_2=2
    b_3\cdot N_3\equiv 1\mod 6 thus, b_3=5

    Thus,
    x\equiv 2400+156+1625\equiv 4181 \mod 390
    Thus,
    x\equiv 281\mod 390
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  3. #3
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    Thanks! That was awesome - this was a problem in m Discrete Math class - so i posted it here! Sorry for trouble caused - am still a bit confused as to how you calculate b1 b2 and b3

    Cheers
    Last edited by darren_a1; April 9th 2006 at 04:37 PM.
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  4. #4
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    Quote Originally Posted by darren_a1
    Thanks! That was awesome - this was a problem in m Discrete Math class - so i posted it here! Sorry for trouble caused - am still a bit confused as to how you calculate b1 b2 and b3

    Cheers
    First, do you understand that those are the numbers that solve the congruences,
    <br />
\left\{ \begin{array}{c}b_1\cdot N_1\equiv 1 \mod 13\\<br />
b_2\cdot N_2\equiv 1\mod 5\\<br />
b_3\cdot N_3\equiv 1\mod 6\end{array}\right<br />
.
    Is your problem based on how to solve these congruences.
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