CAT Quantitative Aptitude Questions | CAT Number Systems - Factors

The question is about Sum of Factors. Given a number, finding the sum of factors is easy. Given the sum of factors, how does one find the number - now that's interesting. 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 1: The sum of the factors of a number is 124. What is the number?

1. Number lies between 40 and 50
2. Number lies between 50 and 60
3. Number lies between 60 and 80
4. More than one such number exists

Explanatory Answer

Method of solving this CAT Question from Number Theory - Factors: In some situations, going back would be really difficult.

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)

For any number N of the form paqbrc, the sum of the factors will be (1 + p¹ + p² + p³ + …+ p^a) (1 + q¹ + q² + q³ + …+ q^b) (1 + r¹ + r² + r³ + …+ r^c).

Sum of factors of number N is 124. 124 can be factorized as 2² * 31. It can be written as 4 * 31, or 2 * 62 or 1 * 124.
2 cannot be written as (1 + p¹ + p² + p³ + …+ p^a) for any value of p.
4 can be written as (1 + 3)
So, we need to see if 31 can be written in that form.
The interesting bit here is that 31 can be written in two different ways.
31 = (1 + 2¹ + 2² + 2³ + 2⁴)
31 = ( 1 + 5 + 5²)
Or, the number N can be 3 * 2⁴ or 3 * 5². Or N can be 48 or 75.

The question is "What is the number?"

Hence the answer is "More than one such number exists"

Choice D is the correct answer.