, 2011

The number $12$ may be factored into three positive integers in exactly eighteen ways, these factorizations include $1 \times 3 \times 4$, $2 \times 2 \times 3$ and $2 \times 3 \times 2$. Let $N$ be the number of seconds in a week. In how many ways can $N$ be factored into three positive integers?

A numerical answer is not sufficient. The calculation should be explained and justified.