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?

AnswerscorrectIy Aug 31, 2024

#1**+1 **

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! :)

NotThatSmart Aug 31, 2024