SOLUTION: 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}

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): You can put this solution on YOUR website!
.

If in the fraction the denominator is relatively prime to the base b,
then this fraction in base b is repeating.

This general rule is ALWAYS valid.

Apply it in your case. The fraction has the denominator 288 relatively prime to the base 13.

THEREFORE, the given fraction in the given base is repeating.

Solved, answered and explained.

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