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avatar+1847 

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?

 Jun 17, 2024
 #1
avatar+1075 
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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 \(\frac{1}{288}\) is in base 25, the decimal is repeating. 

 

So repeating is our answer. 

 

Thanks! :)

 Jun 17, 2024

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