CAT Quantitative Aptitude Questions | CAT Number Systems - Factors

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HCF and LCM Remainders Factorials Digits Ratios,Mixtures;Averages Percents; Profits; SICI Speed & Time; Races Logarithms and Exponents Pipes,Cisterns; Work,Time Set Theory Geometry Coordinate Geometry Mensuration Trigonometry Linear & Quadratic Equations Functions Inequalities Polynomials Progressions Permutation Probability
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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.