Partial set

The partial set of a natural number is the set of all the divisors of this number. It consists of all natural numbers by which the starting number can be divided without a remainder , and is often referred to as or . ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle T_{n}}$ ${\displaystyle T(n)}$

For example, the divisor set of the number 12 consists of all natural numbers by which the 12 can be divided without a remainder, i.e.

• 1 and 12 (because 1 12 = 12, i.e. 12: 1 = 12 and 12: 12 = 1)
• 2 and 6 (because 2 6 = 12)
• 3 and 4 (because 3 4 = 12)

Thus is the divisor ${\displaystyle T_{12}=\{1,2,3,4,6,12\}}$

For the sake of clarity, the divisors are listed here in an orderly manner. The smallest divisor multiplied by the largest gives the examined number, and also the product of the second smallest by the second largest divisor and so on. These pairs of dividers are called complementary dividers .

With the help of the prime factorization , all factors of the subset can be determined quickly, but there are no fast methods for determining the prime factorization.

Formal definition

A natural number is a divisor of a natural number if and only if there is a natural number for which applies. One writes for this formally: ${\displaystyle a}$ ${\displaystyle n}$ ${\displaystyle b}$ ${\displaystyle a\cdot b=n}$

${\displaystyle a\mid n}$.

It goes without saying that it is always ; the numbers and are called complementary factors . ${\displaystyle b\mid n}$ ${\displaystyle a}$ ${\displaystyle b}$

The subset of is ${\displaystyle n}$

${\displaystyle T_{n}=\{d\in \mathbb {N} :d\mid n\}}$.

Number of divisors

How many divisors a number has (in other words, in mathematical terms, the power of its divisor set) cannot be easily seen from this number, but it can be calculated using the prime factorization of the number. This assignment is called the divisor number function . Its first values ​​are 1, 2, 2, 3, 2, 4, 2, ... The properties of this function, especially its behavior for large values ​​of , are treated in number theory . ${\displaystyle n}$

Trivial dividers

Every natural number has at least two factors, namely and . These factors are called the trivial factors . (An exception is the number because the two trivial factors are the same here; it is the only number with only one factor.) ${\displaystyle n}$ ${\displaystyle 1}$ ${\displaystyle n}$ ${\displaystyle 1}$

Natural numbers whose divisors consist of exactly two elements are called prime numbers . If is a prime number then: ${\displaystyle p}$

${\displaystyle T_{p}=\{1,p\}}$

See also

• Divisibility
• Greatest common factor and least common multiple
• Multiple set
• Divisional sum

Web links

• Calculate the divisors of a decimal number with output of the results in decimal, hexadecimal and binary form
• Video: subsets and prime numbers . University of Education Heidelberg (PHHD) 2012, made available by the Technical Information Library (TIB), doi : 10.5446 / 19879 .