lcm of $\frac{2 \pi}{n}$ and $n \pi$

Question

I have been asked

Period of the function $f(x) = \frac{\sin(\sin nx)}{\tan(\frac{x}{n})}, n\in \mathbb{N}$ is $6 \pi$, then what is the value of $n$?

I figured that the period for the numerator should be $\frac{2 \pi}{n}$ and the period for the denominator should be $n \pi$. Now, the lcm of these two periods should be the period of $f$ which is already given, and hence one could easily find out the value of $n$ from that.

But, how do I find out the lcm of $\frac{2 \pi}{n}$ and $n \pi$? I know how to find out lcm for numbers, but this is a little confusing for me and I am getting wrong answers trying it out myself. Please explain.

Answer 1

For your title question, it might be sensible to consider even $n$ and odd $n$ separately.

If $n=2k$ then $\frac{2 \pi}{n} =\frac{ \pi}{k}$ and $n\pi=2k\pi$, where $2k\pi$ is an integer multiple of $\frac{ \pi}{k}$. So the lowest common multiple is $n\pi$

If $n=2k+1 $ then $\frac{2 \pi}{n} =\frac{ 2\pi}{2k+1}$ and $n\pi=(2k+1)\pi$, where $2(2k+1)\pi$ is an integer multiple of $\frac{ 2\pi}{2k+1}$ but $(2k+1)\pi$ is not. So the lowest common multiple is $2n\pi$