But what's a humble number? (1)

A humble number is a number whose prime factors are only $2$ , $3$ , $5$ ,and $7$ .

Thus, in $100!$ , we get the number of factors of the largest humble number that can divide it. That is, if $N$ is the largest humble number that can divide $100!$ , then

$N = 2^{97} \times 3^{48} \times 5^{24} \times 7^{16}$

Each of these exponents were obtained by

$\large \sum_{n=1}^{\infty}\left \lfloor \frac{100}{k^{n}} \right \rfloor$

with $k = 2, 3, 5,$ and $7$ .

Now, the number of factors of $N$ equals

$(97+1) \times (48+1) \times (24+1)\times (16+1) =$ the answer, $2040850$ .