A homothetic arithmetic function of ratio $K$ is a function $f mathbb{N}
ightarrow R$ such that $f(Kn)=f(n)$ for Swabs every $ninmathbb{N}$.Periodic arithmetic funtions are always homothetic, while the converse is not true in general.In this paper we study homothetic and periodic arithmetic functions.In particular we give an upper bound for HERBATINT 4R the number of elements of $f(mathbb{N})$ in terms of the period and the ratio of $f$.