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Math Help - congruence

  1. #1
    Junior Member
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    May 2008
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    55

    Post congruence

    how do i solve this system
    x=3 (mod 5), x= 5 (mod 7), x=7 (mod 11),

    can i please get some help with a step by step process

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  2. #2
    Senior Member JaneBennet's Avatar
    Joined
    Dec 2007
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    First, look at the first two congruences.

    x\equiv3\pmod5\ \Rightarrow\ 7x\equiv21\pmod{35}\ \Rightarrow\ -14x\equiv-42\pmod{35}
    x\equiv5\pmod7\ \Rightarrow\ 5x\equiv25\pmod{35}\ \Rightarrow\ 15x\equiv75\pmod{35}

    Adding gives x\equiv33\pmod{35}\equiv-2\pmod{35}.

    Now compare this with the last congruence.

    x\equiv-2\pmod{35}\ \Rightarrow\ 11x\equiv-22\pmod{385}\ \Rightarrow\ 176x\equiv-352\pmod{385}
    x\equiv7\pmod{11}\ \Rightarrow\ 35x\equiv245\pmod{385}\ \Rightarrow\ -175x\equiv-1225\pmod{385}

    And adding, x\equiv-1577\pmod{385}\equiv348\pmod{385}.

    So the solution set is \{348+385k:k\in\mathbb{Z}\}.
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  3. #3
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    also:

    you may find this helpful:

    here
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