Eventually Equal

You are given three positive integers $A,B,$ and $C$. You would like to make $A$ and $B$ equal to each other. To achieve this, you have the following two types of moves available to you: 1. Simultaneously replace $A$ by $\gcd(A, C)$, and $B$ by $\gcd(B, C)$. Note that both replacements *must* be made: it is not allowed to replace only $A$ or only $B$. 2. Increment $C$ by $1$. Find the **mini