The question is about a number with n factors. Number of factors of N is given. We need to find out the smallest of N. Dealing with factors of a number is a vital component in CAT Number Systems: Factors. A range of CAT questions can be asked based on this simple concept.CAT exam has been testing these type of questions for the past few decades and Factors questions forms an integral part of the CAT syllabus.

Question 10: Find the smallest number that has exactly 18 factors.

- 180
- 216
- 240
- None of these

180

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^{2}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

The question is **"Find the smallest number that has exactly 18 factors."**

Choice C is the correct answer.

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