Number Theory

We can use trial and error to solve this problem. 

First, let's note that

$582_{10} = N7_{25}$

Let's decrease to base 20 and see what happens. 

$582_{10}=192_{20}$

So the number has to be between 20 and 25. 

Let's try 23. 

We have

$582_{10}=127_{23}$

When we convert to 24, we get

$582_{10}=106_{24}$

So the smallest number B must be 25. 

So our answer is 25. 

Thanks! :)