How to find the sum of divisors of a certain number

It will best be illustrated through an example.

1. Find the sum of the divisors of 100.

solution:
First we will go by the conventional method of getting the number of factors of 100 and the add them to get the answer then we will discuss the faster method of getting the required answer.
Now,
          2 x 50 = 100
          4 x 25 = 100
          5 x 20 = 100
          10 x 10 = 100
           1 xx 100 = 100

Therefore, number of factors of 100 are 1,2,5,10,20,25,50,100 and the required sum is 
 1  +2 +4+5 +10+20+25+50+100 = 217

Now the FASTER METHOD.

WRITE 100 in terms of prime factors as 
 100 = 2 x 2 x 5 x 5 = 2²×5² 

Now, the sum of the divisors of 100 is given by = (2⁰ + 2¹ + 2² )(5⁰ + 5¹ + 5²) 
                                                                             = (1+2+4)(1+5+25) 
                                                                             = (7)(31)
                                                                             = 217
Here, is what we have done,
The power of 2 is 2. We calculate 2⁰ , 2¹ ,2² and add them together.
The power of 5 is 2. We calculate 5⁰ , 5¹ ,5² and add them together.
Multiply both the numbers obtained above to get the required answer.