Sum of Divisors

In number theory, many formulas exist that we never think of. The first topic I wrote about number theory is to find the number of divisors an integer has. This time let’s talk about finding the sum of those divisors without listing them individually.


Given a positive integer N with prime factor (a^m)(b^n)(c^p)\ldots the sum of its divisors ( Sn) can be calculated using the formula,

S_n= (\sum_{x=0}^{m}a^m)( \sum_{n=0}^{n}a^n) ( \sum_{x=0}^{n}b^n) ( \sum_{p=0}^{p}c^p)\ldots


Worked Problems 1:

Find the sum of divisors of 12.


12=2^2(3) S_n=(2^0+2^1+2^2)(3^0+3^1) S_n=(7)(4) = 28

It is easy to check that the answer is correct by listing down all the divisors of 12 which is {1,2,3,4,6,12}.


Worked Problem 2:

Find the sum of divisors of 5400.


5400=2^3(3^3)(5^2) S_n=(2^0+2^1+2^2+2^3)(3^0+3^1+3^2+3^3)(5^0+5^1+5^2) S_n=(15)(40)(31) S_n=18,600


Worked Problem 3:

The sum of divisors of an integer N is 7623. N can be written in the form of 2x3y where x and y are all positive integers. What is N?


S_n=(2^0+2^1+2^2+\ldots+2^x)(3^0+3^1+3^2+\ldots+3^y) 7623=(2^0+2^1+2^2+\ldots+2^x)(3^0+3^1+3^2+\ldots+3^y) 7623=(2^{x+1}-1)(\frac{3^{y+1}-1}{2}) 15246=(2^{x+1}-1)( 3^{y+1}-1)


We take note of the following facts;

2^{x+1}-1 is odd and 3^{y+1}-1 is even.

Also 15246=2(3^2)(7)(11^2)

Since 3^{y+1}-1 is even, 2 is one of its factors. And take note that x and y are positive integers. By inspection;

3^{y+1}-1 = 2(11^2) = 242 3^{y+1}= 243 y=4


2^{x+1}-1=3^2(7)=63 2^{x+1}=64 x=5


N=2^x(3^y) N=2^5(3^4) N=2592












Blogger and a Math enthusiast. Has no interest in Mathematics until MMC came. Aside from doing math, he also loves to travel and watch movies.

Latest posts by Dan (see all)

You may also like...

8 Responses

  1. 2xkP2z Really superb information can be found on site.

  2. this site says:

    YphY07 You have brought up a very superb details , regards for the post.

  3. suba me says:

    VbrHfj Regards for this terrific post, I am glad I discovered this web site on yahoo.

  4. 26809 232345You designed some decent points there. I looked online for the concern and identified a lot of people could go as effectively as employing your internet internet site. 616252

  5. 207312 857601Attractive section of content. I just stumbled upon your blog and in accession capital to assert that I acquire actually enjoyed account your blog posts. Anyway I will be subscribing to your augment and even I achievement you access consistently quickly. 50003

  6. 396705 164922I just put the link of your blog on my Facebook Wall. extremely nice blog indeed.,-, 188437

  1. March 24, 2014

    sacoche burberry

    I could not resist commenting. Well written!|

Leave a Reply

Your email address will not be published.