Period of sum of functions
Posted: Wed May 25, 2016 1:44 pm
Let $f:\mathbb{Z} \rightarrow \mathbb{R}$ be a function with period $T>0$, that is $f\left(x+T\right)=f(x)$ forall $x \in \mathbb{Z}$. The least period of a function divides every other period of it.
Let ${\rm a, b}$ be two natural coprime numbers and let $f, g$ be two functions that are onto $\mathbb{Z}$ and have a least period ${\rm a, b}$ respectively. Prove that $f+g$ is periodic and its least period is ${\rm ab}$.
Does the result hold necessary if ${\rm a, b}$ are not coprime? Explain your answer.
Let ${\rm a, b}$ be two natural coprime numbers and let $f, g$ be two functions that are onto $\mathbb{Z}$ and have a least period ${\rm a, b}$ respectively. Prove that $f+g$ is periodic and its least period is ${\rm ab}$.
Does the result hold necessary if ${\rm a, b}$ are not coprime? Explain your answer.