The user-friendly version of this content is available here.

The following content is copyright (c) 2009-2013 by Goods of the Mind, LLC.

This essential trains for: AMC-8, AMC-10, AMC-12, AIME, Math Kangaroo 7-8, Math Kangaroo 9-10.

Suppose we want to calculate the sum of all the positive divisors of an integer.

The fundamental theorem of arithmetic guarantees that a given integer, N has the unique prime factorization:

where pk are ordered prime factors,

and ak are integer exponents greater than or equal to 1:

The set of all the positive divisors of an integer can be found by taking the sum of all the possible products in the table:

After regrouping and factoring, the sum is:

Notice that in each parenthesis there is a geometric sequence. We can sum all the sequences, to obtain:

Notation: The conventional notation for this sum is a function labeled as lowercase "sigma":

Property: The sum of divisors is multiplicative. The product of two integers has the same sum of divisors as the product of the sums of divisors of each number: