First Idea. Wilson's Theorem. But

Wilson's theorem actually states that if

$(n-1)! + 1 \equiv 0 \pmod{n}$

then $n$ is prime. The converse is also true that if $n$ is a prime then $(n-1)! \equiv -1 \pmod{n}$ .

A common observation tells us that in the problem it is stated $7747!$ not $7747$ and $7747! > 7746! + 1$ so the division of $7746!+1$ by $7747!$ is certainly not an integer. Furthermore, wilson's theorem has no application here as $7747 = 61 \times 127$ is a composite number.