+0  
 
-2
3
1
avatar+153 

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.

 Jun 20, 2024

Best Answer 

 #1
avatar+1230 
+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! :)

 Jun 20, 2024
 #1
avatar+1230 
+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

1 Online Users