How to find lcm of $2\pi$, $2\pi/2$, $2\pi/3$ and so on?

Question

In Fourier series, the sum of periodic functions is also a periodic function, period of which is find by taking lcm of periods of all functions. Using this, we can find period of given series: $\sum_{n=1}^{\infty}A_n \cos(nx).$ The answer is that the period of above function is $2\pi$. How?

Answer 1

The smallest positive number that is an integer multiple of $2\pi$, $\frac{2\pi}{2}$, $\frac{2\pi}{3}$, $\frac{2\pi}{4}$, etc. is $2\pi$.

Note that it must be a positive integer multiple of $2\pi$ because it is an integer multiple of all these mubers. So it is at least $2\pi$, and $2\pi$ satisfies because $2\pi = k \cdot \frac{2\pi}{k}$.