Find the unique positive integer $a$ with $a \leq 19$, for which

$$17 ! \cdot 3^{16} \equiv a(\bmod 19) \text {. }$$

Results used should be stated but need not be proved.

Solve the system of simultaneous congruences

$$\begin{aligned} &x \equiv 1(\bmod 2), \\ &x \equiv 1(\bmod 3) \\ &x \equiv 3(\bmod 4) \\ &x \equiv 4(\bmod 5) \end{aligned}$$

Explain very briefly your reasoning.