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avatar+1335 

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 19, 2024
 #2
avatar+1075 
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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! :)

 Jun 19, 2024

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