Does $Ax + By = C$ pass through any lattice point?

Question

Given an equation of a straight line of form $Ax + By = C$. where $A,B,C$ are integers. How could we check if this passes through any lattice point or not?

Please suggest me a suitable algorithm.

Answer 1

Assuming you mean the lattice of integers, that is equivalent to checking if the equation has integer solutions.

Assume that $k\times GCD(A,B)=C$. Then by Euclid's algorithm, there are $X,Y$ such that $AX+BY=GCD(A,B)$, so $AkX+BkY=C$, where $kX$ and $kY$ must be integers.

On the other hand, assume $Ax+By=C$ has a solution. Then $GCD(A,B)\left(\frac{A}{GCD(A,B)}X+\frac{B}{GCD(A,B)}Y\right)=C$ so $GCD(A,B)$ divides $C$.

Answer 2

I assume the lattice you are using is $\mathbb{Z}^2$, consisting of all integer pairs $(x,y)$. What you have is an example of a linear diophantine equation, meaning that you are looking for integer pairs $(x,y)$ satisfying this equation. The key here is to see whether or not $C$ is divisible by $gcd(A,B)$. If so, there are infinitely many solutions (this is known as Bezout's identity).