HMMT February

Each 0 represents a factor of $10=2 \cdot 5$. Thus, we wish to find the smallest factorial that contains at least 290 2's and 290 5's in its prime factorization. Let this number be $n$!, so the factorization of $n$! contains 2 to the power $p$ and 5 to the power $q$, where $$p=\left\lfloor\frac{n}{2}\right\rfloor+\left\lfloor\frac{n}{2^{2}}\right\rfloor+\left\lfloor\frac{n}{2^{3}}\right\rfloor+\cdots \text { and } q=\left\lfloor\frac{n}{5}\right\rfloor+\left\lfloor\frac{n}{5^{2}}\right\rfloor+\left\lfloor\frac{n}{5^{3}}\right\rfloor+\cdots$$ (this takes into account one factor for each single multiple of 2 or 5 that is $\leq n$, an additional factor for each multiple of $2^{2}$ or $5^{2}$, and so on). Naturally, $p \geq q$ because 2 is smaller than 5. Thus, we want to bring $q$ as low to 290 as possible. If $q=\left\lfloor\frac{n}{5}\right\rfloor+\left\lfloor\frac{n}{5^{2}}\right\rfloor+\left\lfloor\frac{n}{5^{3}}\right\rfloor+\cdots$, we form a rough geometric sequence (by taking away the floor function) whose sum is represented by $290 \approx \frac{n / 5}{1-1 / 5}$. Hence we estimate $n=1160$, and this gives us $q=288$. Adding 10 to the value of $n$ gives the necessary two additional factors of 5, and so the answer is 1170.