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avatar+257 

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 $19,$ is it terminating or repeating?

 Jun 24, 2024
 #1
avatar+1230 
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First, let's convert the numbers in to base 19. 

We have

\((F3)_{19} = (15 × 19^1) + (3 × 19^0) = (288)_{10 }\)

 

This means that \(\frac{1}{288}_{10}=\frac{1}{F3}_{19}\)

 

Doing some base division, we have

\(\frac{1}{F3}_{19}=\overline{0.014F9AE6}\)

 

Thus, it is a repeating decimal. 

 

So our answer is repeating. 

 

Thanks! :)

 Jun 24, 2024

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