+617 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?
+1790 First, let's convert 1/288 into base 25 form.
We get that \((BD)_{25} = (11 × 25^1) + (13 × 25^0) = (288)_{10}\)
This means that \(\frac{1}{288}_{10} = \frac{1}{BD}_{25}\)
Doing base division, we have that
\(\frac{1}{BD}_{25} = 0.\overline{0.02468ACEGIKN}\)
So when \(1/288\) is in base 25, the decimal is repeating.
So repeating is our answer.
Thanks! :)
