Two periodic functions $f$ and $g$ have minimum positive periods $2\pi$ and $3\pi$ respectivelly.
Is the sum of these functions periodic? Justify.
My books states the solution is
Yes, because both functions also admit $6\pi$ as period.
Can someone explain this problem to me? Is there any property/identity that I am supposed to use to solve this? I have no clue other than 6 is a multiple of 2 and 3, but I'm not sure if that helps.
$$f(x+6\pi)=f(x+4\pi)=f(x+2\pi)=f(x)$$ $$g(x+6\pi)=g(x+3\pi)=g(x)$$ so $$f(x+6\pi)+g(x+6\pi)=f(x)+g(x)$$
Hint: You are supposed to use the definitions of periodic function and period.
The general principle here is that if something has a period of $T$, it repeats itself every $T$. That means it also repeats itself every $2T$, as well as every $3T$, and so on.
If you have two functions $f$ and $g$ with different fundamental periods, and you want to know if some combination of $f$ and $g$—some function of them, in other words—is also periodic, what you want to do is see if there is some single value that is a multiple of both of those periods.
In this case, with the periods being $2\pi$ and $3\pi$, there is a common multiple of both of those. In fact, there are an infinity of them (as there must be), but the lowest (positive) one is $6\pi$. It does not matter that $6\pi$ itself is irrational; what matters is that it is an integer multiple of the two original periods: It is $3 \times 2\pi$, and it is $2 \times 3\pi$.
We can see this more directly as follows:
LStU's answer shows how to demonstrate this more rigorously. The above is just the intuition for finding the right period to try out.
