Find the smallest positive integer $B$ so that when we express the decimal number $582$ as a base $B$ number, we still get a $2$-digit number.

BRAlNBOLT Jun 20, 2024

#1**+1 **

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! :)

NotThatSmart Jun 20, 2024

#1**+1 **

Best Answer

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! :)

NotThatSmart Jun 20, 2024