Many a times it is asked to find the period of combination of two functions given the period of individual functions.
Let’s take this example:
If $f(x)= \cos ax + \sin x$ is periodic, then $a$ cannot be? $\pi$, 0.3, 0.5, 5
I can see that the period of $\cos ax$ is $\frac{2\pi}{a}$ and that of $\sin x$ is $\pi$ but how to find the period of $f(x)$?
Another example of very similar concept is
If $f(x)$ and $g(x)$ are periodic functions with period $7$ and $11$ respectively. Then the period of $F(x) = f(x) g(x/5) - g(x) f(x/3)$
I know the period of $g(x/5)$ is $11/5$ and that of $f(x/3)$ is $7/3$, but how to combine the periods?
The period of a function is the smallest nonzero $t$ such that
$$f(x+t)=f(x)$$ and by induction $$f(x+kt)=f(x).$$
Now if $f$ is a combination of two functions of known and commensurate periods $t',t''$, there are two relatively prime integers such that
$$k't'=k''t''=t.$$
and we have
$$g(x+k't')=g(x),\\h(x+k''t'')=h(x)$$ which shows that $f$ is certainly periodic with a period not exceeding $t$.
On the other hand, if $t',t''$ are incommensurate, $f$ is aperiodic.
Anyway, the period can be shorter, and this depends on the particular combination of $g,h$ defining $f$. Unfortunately, I know of no way to find the shorter period when there is one, in another way than... looking for the period. E.g. $\sin 5x+\cos x$ and $\sin 5x-\cos x$ are both of period $2\pi$, but their sum $2\sin 5x$ has the period $\dfrac{2\pi}5$.