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

Question

0.\overline{0121}_3 = 0.01210121 \dots,
which is repeating.

When the \frac{1}{288} is expressed in base 15, is it terminating or repeating?

Answer 1

To determine if \frac{1}{288} is terminating or repeating when expressed in base 15, we need to first convert \frac{1}{288} to base 15.

We can start by using long division to convert 1 (base 10) to base 15:

1 ÷ 15 = 0 remainder 1
1 (base 10) = 1 (base 15)

So, 1 (base 10) is 1 (base 15).

Next, we convert 288 (base 10) to base 15:

288 ÷ 15 = 19 remainder 3
19 ÷ 15 = 1 remainder 4
1 ÷ 15 = 0 remainder 1

Therefore, 288 (base 10) is 134 (base 15).

Now, we can express \frac{1}{288} in base 15 as 0.00\underline{0069}23 (where the underlined part repeats), which means that \frac{1}{288} is repeating when expressed in base 15.