3 Products and factorials $\quad$ K

Determine the highest natural power of the number 2007 that divides 2007!
$2007=3^{2} \cdot 223$. In the prime factorization of the number 2007!, the exponent of the number 3 will be quite large, as the factor 3 is included in the factorization of every third number. The factor 223 is only included in the factorization of numbers of the form 223 p, where $p$ is a natural number not exceeding 9. Thus, in the prime factorization of the number 2007!, the number 223 will enter with an exponent of 9. Therefore, the number 2007! will be divisible by $2007^{9}$, but will not be divisible by $2007^{10}$.

## Answer

To the ninth power