I am looking at problems in Vandendriessche and Lee's Problems in elementary number theory and this is one of their problems:
Find $4$ positive integers not exceeding $70000$ such that each have more than $100$ divisors
I just started picking some numbers and looked at their factors:
So I guess $400$ should have $15$ factors based on the pattern above.
In this case I've noticed that there are $3$ more factors after each multiplication by $2$ but I think this approach wouldn't help me to find solutions.