# CAT Quantitative Aptitude Questions | CAT Number Systems - Factors

###### CAT Questions | Number Theory | Smallest Number CAT Questions

The question is about the smallest number with N factors. Number of factors of of a number is given. We need to find out the smallest number. 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 13: What is the smallest number that has exactly 12 factors?

1. 211
2. 60
3. 120
4. 144

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### 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 paqbrc will have (a + 1) (b + 1) (c + 1) factors, where p, q, r are prime.
If a number has exactly 12 factors then (a + 1)(b + 1)(c + 1) should be equal to 12.

If the number has only one prime factor, it should be of the form.
p11 smallest no. possible = 211
If it has 2 prime factors, it can p3q2 or p5q
23 * 32 or 25 * 3
72 or 96
If it has 3 prime factors it can be of the form paqbrc
p2qr = 22 x 3 x 5
p2qr = 60
The smallest number possible = 60

The question is "What is the smallest number that has exactly 12 factors?"

##### Hence the answer is 60.

Choice B is the correct answer.

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