Explanation:

Any number of the form p^aq^br^c will have (a + 1) (b + 1) (c + 1) factors, where p, q, r are prime. (This is a very important idea)
Now, the number we are looking for has 18 factors. It can comprise one prime, two primes or three primes.
Now, 18 can be written as 1 * 18 or 3 * 6 or 9 * 2 or 2 * 3 * 3.
If we take the underlying prime factorization of N to be p^aq^b, then it can be of the form p¹q⁸ or p²q⁵
If we take the underlying prime factorization of N to be p^a, then it can be of the form p¹⁷
If we take the underlying prime factorization of N to be p^aq^br^c, then it can be of the form p¹q²r²
So, N can be of the form p¹⁷, p²q⁵, p¹q⁸ or p¹q²r²
Importantly, these are the only possible prime factorizations that can result in a number having 18 factors.
Now, let us think of the smallest possible number in each scenario
p¹⁷ - Smallest number = 2¹⁷
p²q⁵ – 3² * 2⁵
p¹q⁸ – 3¹ * 2⁸
p¹q²r² – 5¹ * 3² * 2²
The smallest of these numbers is 5¹ * 3² * 2² = 180