Questions on congruences, linear diophantine equations, greatest common divisor, divisibility, etc.

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3
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A natural number $n>2$ is a prime iff $\prod_{k=1}^{n-1} k \equiv n-1 \pmod {\sum_{k=1}^{n-1} k}$

Is this proof acceptable ? Theorem 1 (Wilson). A natural number $n>1$ is a prime iff: $$(n-1)! \equiv -1 \pmod n.$$ Theorem 2. A natural number $n>2$ is a prime iff: ...
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A set of integers whose elements all divide $2015^{200}$ but do not divide each other

Let $S$ be a set of natural numbers,such that each element divides $2015^{200}$ but for no two elements $a$ and $b$, $a|b$. Find the maximum number of elements in $S$ . $2015^{200}=(5\cdot ...