CAT Quantitative Aptitude Questions | CAT Number Systems - Factors

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.

1. 180
2. 216
3. 240
4. None of these

Explanatory Answer

Method of solving this CAT Question from Number Theory - Factors: Given prime factorization, getting to the number of factors is easy. Given number of factors, how does one get to the prime factorization? Chew on that.

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

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

Hence the answer is 180.

Choice C is the correct answer.