If $85_b = 58_c$, what is the smallest possible value of $b$? I have searched and watched videos online and can't find a method to solve this problem:

If $85_b=58_c$ for some positive integer bases $b$ and $c$, what is the smallest possible value for $b$?


$85_b = 58_c$ means $8b + 5 = 5 c + 8$. Thus, $b = (3+5c)/8$. This is an integer whenever $c = 8n + 1$ for some integer $n$. In this case, $b = 5n+1$. Now, since $8$ appears as a digit, we must have $b > 8$. The smallest integer of the form $5n+1$ that is larger than $8$ is $11$.
So the smallest solution is $b = 11$, $c = 17$, as $85_{11} = 58_{17} = 93_{10}$
