Number of factors of a big number

How to find the number of factors of $884466000$ without using a calculator?

Solution 1:

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.