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.
Any number of the form paqbrc 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 paqb, then it can be of the form p1q8 or p2q5
If we take the underlying prime factorization of N to be pa, then it can be of the form p17
If we take the underlying prime factorization of N to be paqbrc, 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."
Choice C is the correct answer.
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