What periodic functions have to do with rational numbers?

Question

I would like to understand why is the answer of this question letter d.

Question:

The solution φ(t) = (x (t) , y(t), z(t), w(t)) of the system of equations:

x' =y, y' = - x, z' =w, w' =-(k^2)*z, that satisfies φ(0) = (1,0,0,1) is periodic.

So, is correct to say:

a) k is a whole number b) k = 0 c) k > 0

d) k is a rational number different of zero

e) k is not a rational number

Answer 1

The solution of this system is made of functions with two distinct periods: $2\pi$ and $2k\pi$. For the solution to be globally periodic, $k$ must be rational, let $\dfrac pq$, so that after $q$ period for the second, you have seen $p$ periods of the first.

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In other words, rational numbers have to do with periodic functions in that the respective periods must be commensurable.