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Math Help - Checking whether a line passes through any lattice points

  1. #1
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    line passes through any points

    Hi, given an equation of a line of the form a\times x+b\times y+c, how can one determine if it passes through any point or not?
    E.g
    2\times x+3\times y=4 passes through some lattice point while
    6\times x+8\times y=25doesn't.
    Thanks.
    Ok, i think i got it, it's a Diopthene equation , and we need to check if c is a multiple of gcd(a,b) right?
    Last edited by pranay; March 30th 2011 at 05:31 PM. Reason: approaching answer
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  2. #2
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    Quote Originally Posted by pranay View Post
    Hi, given an equation of a line of the form a\times x+b\times y+c, how can one determine if it passes through any lattice point or not?
    E.g
    2\times x+3\times y=4 passes through some lattice point while
    6\times x+8\times y=25doesn't.
    Thanks.
    Ok, i think i got it, it's a Diopthene equation , and we need to check if c is a multiple of gcd(a,b) right?
    yes your are correct.

    if \text{gcd}(a,b)|c there are an infinite number of integer solutions
    if \text{gcd}(a,b)\not{|}c there are no integer solutions.
    Thanks from pranay
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