Let $N=p_{1} p_{2} \ldots p_{r}$ be a product of distinct primes, and let $\lambda(N)$ be the least common multiple of $p_{1}-1, p_{2}-1, \ldots, p_{r}-1$. Prove that

$$a^{\lambda(N)} \equiv 1 \bmod N \quad \text { when } \quad(a, N)=1 .$$

Now take $N=7 \times 13 \times 19$, and prove that

$$a^{N-1} \equiv 1 \bmod N \quad \text { when } \quad(a, N)=1 .$$