+1839 When fractions are expressed in different bases, they can be terminating or repeating. For example, when $\frac{1}{5}$ is expressed in base $3,$ the result is
0.\overline{0121}_3 = 0.01210121 \dots,
which is repeating.
When \frac{1}{288} is expressed in base 25, is it terminating or repeating?
+1926 Let's convert the each number in the fraction to base 25.
One always stays the same, so converting 288 to base 25, we have
\((BD)_{25} = (11 × 25^1) + (13 × 25^0) = (288)_{10}\)
This means that
\(\frac{1}{288}_{10} = \frac{1}{BD}_{25}\)
Now, we do base division. We get that
\(\frac{1}{BD}_{25} = 0.\overline{02468ACEGIKN}\)
As you can see, 1/288 in base 25 is repeating.
So repeating is our answer.
Thanks! :)
