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

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?

• @AlexFrancisco I have changed the formatting of the title so as to make it take up less vertical space -- this is a policy to ensure that the scarce space on the main page is distributed evenly over the questions. See here for more information. Please take this into consideration for future questions. Thanks in advance. Mar 19 '18 at 19:52

$288$ 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 $(0.6*10^{-2})_{12} = 0.006_{12}$.

• Perhaps more simply, it can be expressed as $6/12^3=0.006_{12}.$.....+1 Mar 16 '18 at 8:37

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.

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

So it is $0.006_{12}$.