CAT Quantitative Aptitude Questions | CAT Number Systems - Factors

The question is about number of factors. Number of factors of N² is given. We need to find out the number of factors 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. Make use of 2IIMs Free CAT Questions, provided with detailed solutions and Video explanations to obtain a wonderful CAT score.

Question 11: A number N² has 15 factors. How many factors can N have?

1. 5 or 7 factors
2. 6 or 8 factors
3. 4 or 6 factors
4. 9 or 8 factors

Explanatory Answer

Method of solving this CAT Question from Number Theory - Factors: Number has 15 factors, how many factors might square root of that number have. These open-ended questions are very instructive.

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)

N² has 15 factors.
Now, 15 can be written as 1 * 15 or 3 * 5.
If we take the underlying prime factorization of N² to be paqb, then it should have (a + 1) (b + 1) factors. So, N can be of the form
p¹⁴ or p²q⁴
p¹⁴ will have (14 + 1) = 15 factors
p²q⁴ will have (2 + 1) * (4 + 1) = 15 factors. Importantly, these are the only two possible prime factorizations that can result in a number having 15 factors.

The question is "How many factors can N have?"

Hence the answer is 6 or 8 factors.

Choice B is the correct answer.