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Math Help - Need to check my answer

  1. #1
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    Question Need to check my answer

    Thanks for showing me how to work out some of the previous post.

    I would like to check my answer for this problem

    Find all incongruent solutions modulo 35 of the following system

    7x + 5y = 13 (mod 35)
    11x + 13y = 4 (mod35)


    91x + 65y = 169 (mod 35)
    -605x - 65y = -20 (mod 35)
    =
    514 = 149 (mod 35)

    Euclidean Algorithm
    514 = 14*35 + 24

    35 = 1*24 + 11

    24 = 2*11 + 2

    11 = 5*2 + 1

    Extended Euclidean Algorithm
    1 = (1 * 11) + (-5 * 2)
    = (-5 * 24) + (11 * 11)
    = (11 * 35) + (-16 * 24)
    = (-16 * 514) + (235 * 35)
    = (235 * 35) + (-16 * 514)

    35-4 = 31
    Solution
    x = 31 (mod 35)


    77x +44y = 143(mod 35)
    -77x -91y = -28 (mod 35)
    =
    36y = 115 (mod 35)

    Euclidean Algorithm
    35 = 0*36 + 35

    36 = 1*35 + 1

    35 = 35*1 + 0

    Extended Euclidean Algorithm
    1 = (1 * 36) + (-1 * 35)
    = (-1 * 35) + (1 * 36)


    115*36 = 4140
    4140 / 35 = 118.28
    118 * 35 = 4130
    4140-4130 = 10

    Solution
    Y = 10 (mod 35)


    I think im correct but not sure
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  2. #2
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    Quote Originally Posted by math_cali View Post
    7x + 5y = 13 (mod 35)
    11x + 13y = 4 (mod35)
    Because mathematicians are lazy I will not write "mod 35" rather I will write, \equiv and you will know it is modulo 35.

    Now, we have.
    \left\{ \begin{array}{c}7x+5y\equiv 13\\ 11x+13y\equiv 4 \end{array} \right\}
    Multiply the first equation by 13, second by -5,
    \left\{ \begin{array}{c}21x+30y\equiv 29 \\ 15x+5y\equiv 15 \end{array} \right\}
    Note I reduced everything to its smallest positive integer.
    Add them, (note 35y\equiv 0)
    x\equiv 9
    Put that into any equation, say the first.
    7(9)+5y\equiv 13
    28+5y\equiv 13
    5y\equiv 20
    Since,
    \gcd(5,35)=5
    And divides 20.
    We will have 5 incongruent solutions.
    You should be able to solve those.
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