+219 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.
+1926 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! :)
+1926 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! :)
