How do I find the smallest positive integer that can be written as a linear combination of 2191 and 1351?

Question

This is one of my assigned questions, but I don't understand how to answer it as there was nothing like it in any of the example questions. I'm able to write the $gcd(2191, 1351)$ as a linear combination in the form $7 = 37(2191) - 60(1351)$ but I don't know where to go from there.

Answer 1

In general, the smallest positive integer that can be written as an integer linear combination of $a$ and $b$ is the greatest common divisor of $a$ and $b$. This is a consequence of Bézout's lemma. Since you have determined that $\gcd(2191,1351)=7$, you have already answered your question.