How to determine the fundamental period of this discrete time function?

Question

How to determine the fundamental period of this discrete time function? $$x[n]=cos(0.6\pi n)$$

To find a period, I do $$\cos(0.6\pi n)=(0.6\pi n + 0.6\pi N),$$ where both $n$ and $N$ have to be integers.

Now, you need to have $$0.6\pi N=2\pi k.$$ So you have $$N=\frac{2}{0.6}k,$$ and $k$ has to be an integer. What is the fundamental period and why?

Answer 1

Note that $$N = \frac{2}{0.6}k=\frac{10}{3}k.$$ Since both $N$ and $k$ are integers, $k$ must be a multiple of $3$. Hence, the smallest positive integer of the form $\frac{10}{3}k$ is $$N = \frac{10\times 3}{3}=10.$$