Computing $\mathrm{gcd} (100!, 3^{100})$

Question

I am trying to compute $\mathrm{gcd}(100!,3^{100})$. I am still not sure how to reach an answer but I feel that Wilson's Theorem (i.e., $(p-1)!\equiv -1 \bmod p, p$ prime) and Fermat's Little theorem play a key role in the development of the solution.

Answer 1

What power of $3$ divides $100!$?

$\left\lfloor\frac{100}{3}\right\rfloor$ for all the multiples of $3$ in $\{1,2,3\ldots,100\}$.

Additionally, $\left\lfloor\frac{100}{9}\right\rfloor$ for all the multiples of $9$.

Additionally, $\left\lfloor\frac{100}{27}\right\rfloor$ for all the multiples of $27$.

Additionally, $\left\lfloor\frac{100}{81}\right\rfloor$ for all the multiples of $81$.