How to calculate smallest number with certain number of divisors?

Question

From Project Euler problem 500

The number of divisors of 120 is 16. In fact 120 is the smallest number having 16 divisors.

Find the smalle

Answer 1

The highest divisor any number has, other than itself, is the half of the number. For example, 120 has a max divisor of 60 other than itself. So, you can easily reduce the range from (n+1) to (n/2).

More over, for a number to have m divisors, the number must be atleast ((m-1) * 2) following the above logic (-1 because the m th number is itself). For example, a number with 4 divisors has to be atleast 6. So, your search for n has a smaller range now.

These two will reduce the runtime a little.