Suppose $f(x)$ is periodic with period p and $g(x)$ is periodic with period q. Let $r$ be the L.C.M. of p and q, if it exists. Then show that:
If $f(x)$ and $g(x)$ cannot be interchanged by adding a least positive number $k<r$, to $x$, then $r$ is the period of $f(x)+g(x)$.
If $f(x)$ and $g(x)$ can be interchanged by adding a least positive number $k<r$, to $x$, then $k$ is the period of $f(x)+g(x)$.
I know I should give my approach towards the problem but I can't seem to be able to make the head or tail of this problem. I don't think I even completely understand what's being said.
We are given that $f(x+p)=f(x)$ and $g(x+q)=g(x)$. If $r = \operatorname{LCM}(p,q)$, then for sure, we know that $f(x+r)+g(x+q)=f(x)+g(x)$, so $f+g$ is "periodic" with period $r$, but I put "periodic" in quotation marks because it is possible that $f+g$ has a period smaller than $r$. The problem is asking you to show that this will happen if and only if you can "interchange" $f$ and $g$ by "adding $k$ to $x$."
For example, if $f(x+k)=g(x)$ and $g(x+k)=f(x)$, then we have $f(x+k)+g(x+k) = g(x)+f(x)$, so $f+g$ has period $k$. Hope this makes things clearer!