# CAT Practice : Number System: Factors

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Given a number, finding the sum of factors is easy. Given the sum of factors, how does one find the number - now thats interesting.

## Number Theory - Factors

Q.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

Choice D. More than one such number exists.

## Detailed Solution

Any number of the form paqbrc 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 + …+ pa) (1 + q1 + q2 + q3 + …+ qb) (1 + r1 + r2 + r3 + …+ rc).

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…pa) 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.

Correct Answer: More than one such number exists.

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## More questions from Number System - Factors

Dealing with factors of a number is a vital component in this topic. A range of questions can be asked based on this simple concept.