1. Determine the smallest non-zero natural number $n$ for which $\frac{1 \cdot 2 \cdot 3 \cdot \ldots \cdot n}{6^{6}}$ is a natural number.

SGM 2014
Since $6^{6}=2^{6} \cdot 3^{6}$, the number $\frac{1 \cdot 2 \cdot 3 \cdot \ldots \cdot n}{6^{6}}$ is natural if $\quad 1 p$ p

$2^{6} \mid 1 \cdot 2 \cdot 3 \cdot \ldots \cdot n$, respectively $\quad 3^{6} \mid 1 \cdot 2 \cdot 3 \cdot \ldots \cdot n \quad 2 p$

Completion $n=15$ 3p