Minimize $152207x-81103y$ over the positive integers.

Question

Minimize the expression $152207x-81103y$ over the positive integers,
given $x,y\in\mathbb{Z}.$

So the book takes me through modular arithmetic and how to find $\text{gcd}(a,b)$ in order to solve diophantine equations. Then this question pops up in the same chapter.

I know how calculate using modular arithmetic, I know how to find $\text{gcd}(a,b)$ and solve diophantine equations but I don't know how to bunch them up together in order to solve this.

How should I think?

Answer 1

Since $\gcd (152207,81103)=1111$ it is the same as minimum of $$1111(137x-73y)$$

Since $137x-73y=1$ is solvable (say $x=8$ and $y=15$) the answer is $1111$.