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Math Help - GCD equation question

  1. #1
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    GCD equation question

    The following question was confusing me so any help would be appreciated!

    Find the solution in integers to the equation
    155x+341y+385z=1 (where 1 is the gcd of 155,341, and 385).
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  2. #2
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    Quote Originally Posted by clockingly View Post
    The following question was confusing me so any help would be appreciated!

    Find the solution in integers to the equation
    155x+341y+385z=1 (where 1 is the gcd of 155,341, and 385).
    Okay, rewrite this equation as,
    155x+341y = 1 - 385 z \ \ \ (1).
    Now \gcd(155,341)=31 this means (1) has a solution whenever 31 | (1-385z), i.e. 385z\equiv 1 (\bmod 35).
    But this congruence has no solution to it because 385 \equiv 0 under this modulos. Thus, this equation has no solutions.
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