The $n$-factorial of a positive integer $x$ is the product of all positive integers less than or equal to $z$ that are congruent to $z$ modulo $n$. For example, for the number 16, its 2-factorial is $16 \times 14 \times 12 \times 10 \times 8 \times 6 \times 4 \times 2$, its 3-factorial is $16 \times 13 \times 10 \times 7 \times 4 \times 1$ and its 18-factorial is 16. A positive integer is called olympic if it has $n$ digits, all different than zero, and if it is equal to the sum of the $n$-factorials of its digits. Find all positive olympic integers.