Find the largest divisor of 6006006006 that does not exceed 60,000 factoring it, we have
$6 * 1001*1000001 = 2*3*7*11*13* (100^3+1) = 2*3*7*11*13*101*9901$
with this prime factorization, how do you check the largest divisor without too much guessing and checking?

One way is to look for targets to get close to and just use your own numerical nous. Here, the obvious targets are the given $60000$, and $6006006006/60000 \approx 100100$.
$2\cdot3\cdot9901=59406$.
For this instance, start with $60000=10000\cdot6$ and notice that $9901$ is a little less than $10000$, then look for $6$.
With a more complicated factorization it's worth checking quite a few options.
