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Question 1207598: 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 $13,$ is it terminating or repeating?
Found 2 solutions by greenestamps, ikleyn:
Answer by greenestamps(13200)   (Show Source):
You can put this solution on YOUR website!
The fraction 1/n, when expressed in base b, will terminate if and only if the prime factorization of n contains only factors that are prime factors of the base b.
For example, in our familiar base 10 system, since the prime factorization of 10 is 2*5, a fraction 1/n will terminate if and only if the prime factorization of n contains only 2's and 5's.
In your example, the base 13 is a prime number, so 1/n expressed in base 13 will terminate if and only if the prime factorization of n contains only 13's -- i.e., if and only if n is a power of 13.
288 is not a power of 13, so 1/288 expressed in base 13 will not terminate.
ANSWER: Repeating
Answer by ikleyn(52781)  (Show Source):
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