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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: