Numbers and Sets | Part IA, 2005

Find the unique positive integer aa with a19a \leq 19, for which

17!316a(mod19)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

x1(mod2),x1(mod3)x3(mod4)x4(mod5)\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.

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