When the fraction $\frac 1{288}$ is expressed in base $12$, is it terminating or repeating?

Question

I am a student (middle school, so I would be very appreciative if you used simple terms), and I got stumped on this problem:

When the fraction $\dfrac{1}{288}$ is expressed in base $12$, is it terminating or repeating?

Answer 1

$288$ is equal to $2^{5}3^2$. $2$ and $3$ are both factors of $12$, so $\frac1{288}_{10}$ in base-12 terminates. In particular, it can be expressed as $\frac12 12^{-2}$, or $(0.6*10^{-2})_{12} = 0.006_{12}$.

Answer 2

$\displaystyle \frac{1}{288}=\frac{0}{12}+\frac{0}{12^2}+\frac{6}{12^3}$.

So it is $0.006_{12}$.

Answer 3

Note that

$$\displaystyle \frac{1}{288}=\frac{6}{1728}=6\times 12^{-3}$$

Thus its representation in base 12 is $$0.006$$

Which is terminating.