19. Let $p_{1}, \cdots, p_{r}$ be pairwise distinct primes, $m=p_{1}^{k_{1}} \cdots p_{r}^{k_{r}}$, $k_{j}(1 \leqslant j \leqslant r)$ be positive integers, and $\lambda(m)=\left[\varphi\left(p_{1}^{k_{1}}\right), \cdots, \varphi\left(p_{r}^{k_{r}}\right)\right]=\left[p_{1}^{k_{1}-1}\left(p_{1}-1\right), \cdots\right.$, $\left.p_{r}^{k_{r}-1}\left(p_{r}-1\right)\right]$ (see the previous problem). Prove: when $(a, m)=1$, we have $m \mid a^{\lambda(m)}-1$. This provides another proof of Example 5(i) in §4, and also proves the first part of Exercise 28 in §3 when $(a, m)=1$.