Define $\displaystyle f: \mathbb{N} \longrightarrow \mathbb{N}$ by $\displaystyle f(1)=1$ and $\displaystyle f(n)=\sum_{j=1}^k p_j^{r_j},$ where $\displaystyle \prod_{j=1}^k p_j^{r_j}$ is the prime factorization of $\displaystyle n.$ Prove that for any positive integers $\displaystyle x_1, \cdots , x_m$ we have $\displaystyle f(\text{lcm}\{x_1, \cdots , x_m\}) \leq \sum_{j=1}^m x_j.$

Source: American Mathematical Monthly