Sum of Divisors of a Composite Number

Understanding the sum of divisors of a composite number is an important concept in number theory, which has applications in various fields such as cryptography, computer science, and pure mathematics. A composite number is a positive integer that has at least one divisor other than 1 and itself. To explore this concept, we will delve into the definition, properties, and methods for finding the sum of divisors of a composite number.

Definition

The sum of divisors of a number is the sum of all positive divisors of that number, including 1 and the number itself. For a composite number ( n ), this sum is denoted by ( \sigma(n) ).

Properties

The sum of divisors function, ( \sigma(n) ), has several important properties:

• Multiplicative: If two numbers ( a ) and ( b ) are coprime (i.e., their greatest common divisor is 1), then ( \sigma(ab) = \sigma(a) \cdot \sigma(b) ).
• Divisor Summation: For a prime number ( p ) raised to the power ( k ), the sum of divisors is given by ( \sigma(p^k) = 1 + p + p^2 + \ldots + p^k ).

Formula

The sum of divisors of a composite number can be calculated using the following formula:

[ \sigma(n) = \prod_{i=1}^{r} \frac{p_i^{k_i + 1} - 1}{p_i - 1} ]

where ( n ) is the composite number expressed as a product of its prime factors ( p_i ) raised to their respective powers ( k_i ), and ( r ) is the number of distinct prime factors of ( n ).

Table of Differences and Important Points

Property	Description
Multiplicative	( \sigma(ab) = \sigma(a) \cdot \sigma(b) ) if ( a ) and ( b ) are coprime
Divisor Summation	For a prime ( p ) and integer ( k ), ( \sigma(p^k) = \frac{p^{k+1} - 1}{p - 1} )
Formula	( \sigma(n) = \prod_{i=1}^{r} \frac{p_i^{k_i + 1} - 1}{p_i - 1} ) for ( n = p_1^{k_1} \cdot p_2^{k_2} \cdots p_r^{k_r} )

Examples

Let's go through some examples to illustrate how to calculate the sum of divisors of a composite number.

Example 1: Sum of Divisors of 12

The number 12 is a composite number with prime factorization ( 12 = 2^2 \cdot 3^1 ). Using the formula, we get:

[ \sigma(12) = \left(\frac{2^{2+1} - 1}{2 - 1}\right) \cdot \left(\frac{3^{1+1} - 1}{3 - 1}\right) = (7) \cdot (4) = 28 ]

So, the sum of divisors of 12 is 28.

Example 2: Sum of Divisors of 30

The number 30 is a composite number with prime factorization ( 30 = 2^1 \cdot 3^1 \cdot 5^1 ). Using the formula, we get:

[ \sigma(30) = \left(\frac{2^{1+1} - 1}{2 - 1}\right) \cdot \left(\frac{3^{1+1} - 1}{3 - 1}\right) \cdot \left(\frac{5^{1+1} - 1}{5 - 1}\right) = (3) \cdot (4) \cdot (6) = 72 ]

So, the sum of divisors of 30 is 72.

Example 3: Sum of Divisors of 45

The number 45 is a composite number with prime factorization ( 45 = 3^2 \cdot 5^1 ). Using the formula, we get:

[ \sigma(45) = \left(\frac{3^{2+1} - 1}{3 - 1}\right) \cdot \left(\frac{5^{1+1} - 1}{5 - 1}\right) = (13) \cdot (6) = 78 ]

So, the sum of divisors of 45 is 78.

Conclusion

The sum of divisors of a composite number can be calculated using a formula that involves the prime factorization of the number. This concept is not only theoretically interesting but also has practical applications in various areas of mathematics and computer science. Understanding the properties and methods to calculate the sum of divisors is essential for students and professionals working with number theory.