<s>[INST] Number of factors of a big number How to find the number of factors of $884466000$ without using a calculator?
 [/INST] It's not too difficult to factor, since it's divisible by rather high powers of $2$ and $5$; right off the bat, we see that
$$884466000 = 1000 \cdot 884466 = 1000 \cdot 2 \cdot 442233$$
Now $442233$ has digit sum $4 + 4 + 2 + 2 + 3 + 3 = 18$, so it's divisible by $9$, giving
$$442233 = 49137 \cdot 9 = 16379 \cdot 27$$
Looking back at the original form of the number, it's also clearly divisible by $11$, since $$442233 = 11 \cdot 40203$$
So summarizing what we've got so far,
$$884466000 = 2^4 5^3 3^3 11^1 \cdot 1489$$
Now finally, $\sqrt{1489} < 40$, and checking the primes between $13$ and $37$, we find no more divisors. Hence $1489$ is prime.

Thus, every factor of $884466000$ is of the form
$$2^a 3^b 5^c 11^d 1489^e$$
with $0 \le a \le 4$, $0 \le b, c, \le 3$, and $0 \le d, e\le 1$, for a total of $5 \cdot 4^2 \cdot 2^2$ divisors.
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