SCHWEITZER 1985 Problem 7

Let $p_1$ and $p_2$ be positive real numbers. Prove that there exist functions $f_i\colon \mathbb R \rightarrow \mathbb R$ such that the smallest positive period of $f_i$ is $p_i (i=1, 2)$ , and $f_1-f_2$ is also periodic. [J. Riman]