For prime factorization, is there another way of doing it, distinct from dividing the number by a series of primes (starting by the smallest)?
Couldn't we also pick the same series of primes and multiply them somehow until we got the target number?
It's clear that any approach will imply lots of computation, but some ways could be tougher than others, couldn't they?
Is there a name for an exploratory multiplication to try to reach a number?
I am specially interested in doing this with big numbers. It's clear that finding the prime factors of 24 or 32 is an easy task in both directions (dividing or multiplying). But with things like 598703019332, would it be feasible at all?
"Is there a name for an exploratory multiplication to try to reach a number?"
Yes, it is called integer factoring, and it has received a lot of attention since ancient times, but even more so in the last few decades since it has become of practical significance. Just Google it, you will find more information about it than you probably want.