What is wrong with the following argument?
$10^5−1 \not \equiv 1 \pmod 5$ implies 5 is composite.
FlT says:
$$ ( n \text{ prime } \land n \not \mid a ) \quad \implies \quad a^{n-1} \equiv 1 \pmod n $$
The contrapositive says
$$ a^{n-1} \not \equiv 1 \pmod n \quad \implies \quad (n \text{ composite } \textcolor{red}{ \lor n \mid a} ) $$
So the given statement is incomplete, and should be:
$10^5−1 \not \equiv 1 \pmod 5$ implies 5 is composite or 5 divides 10.