The sum of the factors of a number is 124. What is the number?
Choose the correct choice from the following:	A) Number lies between 40 and 50	B) Number lies between 50 and 60	C) Number lies between 60 and 80	D) More than one such number exists	E) None of these
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 . Option More than one such number exists
<result>D</result>
The sum of the factors of a number is 124. What is the number?