To find the amount of factors a number has:
1) Use prime decomposition
Example: 36 = 2^2 * 3^2
2) Add one to each power and then multiply them. This is because the amount of arrangements possible with its prime factors determines the amount of factors. Add 1 because 1 is a possible multiple (2^0 = 1)
Example:
2^2 -> 3
3^2 -> 3
3 * 3 = 9
36 has 9 factors
Reversing this process, we can find the smallest number with n factors:
1) Prime decompose n (n = 36)
Example: 36 = 2^2 * 3^2
2) Arrange the primes in a list in order
Example: 2, 2, 3, 3
3) Subtract one from each
Example: 1, 1, 2, 2
4) Reverse the list
Example: 2, 2, 1, 1
5) Write down the primes in ascending order with the list numbers as the powers
Example: 2^2 * 3^2 * 5 * 7 = 1260
1260 is the smallest number with 36 divisors.
You can of course arrange the prime factors in any order with powers of 1, 1, 2 and 2 to get a number with 36 factors.
Thanks for reading my poor explanation :D