Any number of the form p^{a}q^{b}r^{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^{a}q^{b}, then it can be of the form
p^{1}q^{8} or p^{2}q^{5}
If we take the underlying prime factorization of N to be p^{a}, then it can be of the form
p^{17}
If we take the underlying prime factorization of N to be p^{a}q^{b}r^{c}, then it can be of the form
p^{1}q^{1}r^{2}
So, N can be of the form p^{17}, p^{2}q^{5}, p^{1}q^{8} or p^{1}q^{2}r^{2}
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
p^{17} - Smallest number = 2^{17}
p^{2}q^{5} – 3^{2} * 2^{5}
p^{1}q^{8} – 3^{1} * 2^{8}
p^{1}q^{2}r^{2} – 5^{1} * 3^{2} * 2^{2}
The smallest of these numbers is 5^{1} * 3^{2} * 2^{2} = 180
Correct Answer: 180
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