CAT Number Theory - Interesting question

This is an interesting question from Number Theory. Slightly unconventional, but interesting nevertheless.

Question

The sum of the factors of a number is 124. What is the number?

Correct Answer

Number could be 48 or 75

Explanatory Answer
This video gives the solution for this question. Given below the video is the explanation (in words)

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 p^aq^br^c, 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 …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.