MUHAMMAD YUSRAN BASRI
1211441023
MATH ICP B 2012
UNIVERSITAS NEGERI MAKASSAR
SUMMARY
Definition
For a positive integer n denote by σ(n) the sum of its positive divisors, including 1 and n itself. It
is clear that
𝜎 = ∑𝑑
𝑑[𝑛
This representation will help us to show that 𝜎 is multiplicatiave.
Proposition
If n =𝑝1𝑎1 …..𝑝𝑘𝑎𝑘 Is the prime factorization of n , then
Proof of the proposition
The divisors of n can be written in the form
𝑝1𝑎1 …..𝑝𝑘𝑎𝑘
Where a1…ak
are integers with 0≤ 𝑎1 ≤ 𝛼1…..0≤ 𝑎𝑘 ≤ 𝛼𝑘
Each divisor of n appears exactly once as a summand in the expansion of the product
From which the desired result follows, by also noting the formula for the sum of a finite
geometric progression:
Example
Find the sum of even positive divisors of 100000.
Solution :
The even divisors of 10000 can be written in the form of 2a5b where a and b are integers with
1 ≤ 𝑎 ≤ 5, … ,0 ≤ 𝑏 ≤ 5. Each even divisors of 10000 appears exactly once as a summand in the
expansions of the product