# Thread: Checking whether a line passes through any lattice points

1. ## line passes through any points

Hi, given an equation of a line of the form $\displaystyle a\times x+b\times y+c$, how can one determine if it passes through any point or not?
E.g
$\displaystyle 2\times x+3\times y=4$ passes through some lattice point while
$\displaystyle 6\times x+8\times y=25$doesn'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?

2. Originally Posted by pranay
Hi, given an equation of a line of the form $\displaystyle a\times x+b\times y+c$, how can one determine if it passes through any lattice point or not?
E.g
$\displaystyle 2\times x+3\times y=4$ passes through some lattice point while
$\displaystyle 6\times x+8\times y=25$doesn'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?
if $\displaystyle \text{gcd}(a,b)|c$ there are an infinite number of integer solutions
if $\displaystyle \text{gcd}(a,b)\not{|}c$ there are no integer solutions.