Let $a$ and $b$ be positive integers that are relatively prime, and let $A=8a+3b$ and $B=3a+2b$. The greatest common divisor of $A$ and $B$ is not 1. What is the greatest common divisor of $A$ and $B$?
Solution. Let $d$ be the greatest common divisor of $A$ and $B$. If $d \mid A$ and $d \mid B$, then $d \mid 8B - 3A = 7b$, and $d \mid 2A - 3B = 7a$.

We have found that $d$ is a common divisor of $7a$ and $7b$. Considering that $a$ and $b$ are coprime, the greatest common divisor of $7a$ and $7b$ is 7. (This follows from the fundamental theorem of arithmetic, which states that every natural number can be uniquely expressed as a product of prime numbers, disregarding the order of the factors.)

Accordingly, $d$ is a divisor of 7, so $d=1$ or $d=7$. We know that $A$ and $B$ are not coprime, i.e., $(A; B) \neq 1$, from which it follows that $(A; B)=7$.