factors and number

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


Answer:

Notes:

If a number N can be expressed as product of prime factors such that

N = p^a x q^b x r^c...............................

Write the power of each of the prime factors beginning with zero and going to thee highest power of the prime number.

Record the same for each prime number.

An example will help

Find the total number of factors of 222.

222 = 2 x 3 x 37

Or, 222 = 2¹ x 3¹ x 37¹

then the sum of factors are (2⁰ + 2¹) (3⁰ + 3¹) (37⁰ + 37¹) = (1+2)(1+3)(1+37) = 3 x 4 x 38 = 456

now let us come to the question:
124 can be written as 2 x 62, 4 x 31

this can be written as (2⁰ + 2¹) (62) or (3⁰ + 3¹)(31)
clearly the first option (2⁰ + 2¹) (62) is not possible because 3 x 62 is  not equal to 124
consider (3⁰ + 3¹)(31)
31 can be expressed as (2⁰ + 2¹ + 2² + 2³ + 2⁴ )
31 can be expressed as (5⁰ + 5¹ + 5² )

So, the numbers would be  3¹ x 2⁴  = 48 or  3¹ x 5² = 75