HMMT February

Number theory Difficulty 5.0 AIME Find the answer

Compute the largest positive integer such that $\frac{2007!}{2007^{n}}$ is an integer.

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Note that $2007=3^{2} \cdot 223$ . Using the fact that the number of times a prime $p$ divides $n!$ is given by $\left\lfloor\frac{n}{p}\right\rfloor+\left\lfloor\frac{n}{p^{2}}\right\rfloor+\left\lfloor\frac{n}{p^{3}}\right\rfloor+\cdots$ it follows that the answer is 9.

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