Question 14: Trailing Zeroes

To determine the number of trailing zeroes in base $b$ of given integer $n!$ , we to need find the greatest prime factor of given base. Since prime factors of $6=2\times 3$ . Now we can perform $T_z(n) = \sum_{k=1}^{\infty} \left\lfloor \dfrac{n}{3^k}\right\rfloor \\ \therefore T_z(2188) =\left\lfloor \dfrac{2188}{3}\right\rfloor + \left\lfloor \dfrac{2188}{3^2}\right\rfloor +\left\lfloor \dfrac{2188}{3^3}\right\rfloor +\cdots \\ T_z(2188) = 729 + 243 + 81 + 27 +9+3+1+ 0 + \cdots \\ T_z(2188) = \boxed{1093}$

Note: It is also important to note down the total number of $2's$ in given $n$ .