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The positive integer divisors of 24 are: 1, 2, 3, 4, 6, 8, 12, and 24.

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In mathematics, and specifically in number theory, a

**divisor function** is an arithmetic function related to the divisors of an integer. When referred to as

*the* divisor function, it counts the

*number of divisors of an integer*. It appears in a number of remarkable identities, including relationships on the Riemann zeta function and the Eisenstein series of modular forms. Divisor functions were studied by Ramanujan, who gave a number of important congruences and identities.
A related function is the divisor summatory function, which, as the name implies, is a sum over the divisor function.
The

**sum of positive divisors function** σ

_{x}(

*n*), for a real or complex number

*x*, is defined as the sum of the

*x*th powers of the positive divisors of

*n*. It can be expressed in sigma notation as
where

is shorthand for "

*d* divides

*n*". The notations

*d*(

*n*), ν(

*n*) and τ(

*n*) (for the German

*Teiler* = divisors) are also used to denote σ

_{0}(

*n*), or the

**number-of-divisors function** (sequence in OEIS). When

*x* is 1, the function is called the

**sigma function** or

**sum-of-divisors function**, and the subscript is often omitted, so σ(

*n*) is equivalent to σ

_{1}(

*n*) ().
The

**aliquot sum** s(n) of

*n* is the sum of the proper divisors (that is, the divisors excluding

*n* itself, ), and equals σ

_{1}(

*n*) −

*n*; the aliquot sequence of

*n* is formed by repeatedly applying the aliquot sum function.
For example, σ

_{0}(12) is the number of the divisors of 12:
while σ

_{1}(12) is the sum of all the divisors:
and the aliquot sum s(12) of proper divisors is:
The cases , and so on are tabulated in , , , , , ...
For a non-square integer every divisor d of n is paired with divisor n/d of n and

is then even; for a square integer one divisor (namely

) is not paired with a distinct divisor and

is then odd.
For a prime number

*p*,
because by definition, the factors of a prime number are 1 and itself. Also,where

*p*_{n}# denotes the primorial,
since

*n* prime factors allow a sequence of binary selection (

or 1) from

*n* terms for each proper divisor formed.
Clearly,

and σ(

*n*) >

*n* for all

*n* > 2.
The divisor function is multiplicative, but not completely multiplicative. The consequence of this is that, if we write
where

*r* =

*ω*(

*n*) is the number of distinct prime factors of

*n*,

*p*_{i} is the

*i*th prime factor, and

*a*_{i} is the maximum power of

*p*_{i} by which

*n* is divisible, then we have
which is equivalent to the useful formula:
It follows (by setting

*x* = 0) that

*d*(

*n*) is:
For example, if

*n* is 24, there are two prime factors (

*p*_{1} is 2;

*p*_{2} is 3); noting that 24 is the product of 23×31,

*a*_{1} is 3 and

*a*_{2} is 1. Thus we can calculate

as so:
The eight divisors counted by this formula are 1, 2, 4, 8, 3, 6, 12, and 24.
We also note

*s*(

*n*) =

*σ*(

*n*) −

*n*. Here

*s*(

*n*) denotes the sum of the proper divisors of

*n*, i.e. the divisors of

*n* excluding

*n* itself. This function is the one used to recognize perfect numbers which are the

*n* for which

*s*(

*n*) =

*n*. If

*s*(

*n*) >

*n* then

*n* is an abundant number and if

*s*(

*n*) <

*n* then

*n* is a deficient number.
If n is a power of 2, e.g.

, then

and

*s(n) = n - 1*, which makes

*n* almost-perfect.
As an example, for two distinct primes

*p* and

*q* with

*p < q*, let
Then
and
where

*φ*(

*n*) is Euler's totient function.
Then, the roots of:
allows us to express

*p* and

*q* in terms of

*σ*(

*n*) and

*φ*(

*n*) only, without even knowing

*n* or

*p+q*, as:
Also, knowing n and either

*σ*(

*n*) or

*φ*(

*n*) (or knowing p+q and either

*σ*(

*n*) or

*φ*(

*n*)) allows us to easily find

*p* and

*q*.
In 1984, Roger Heath-Brown proved that
will occur infinitely often.
Two Dirichlet series involving the divisor function are:
which for

*d*(

*n*) =

*σ*_{0}(

*n*) gives
and
A Lambert series involving the divisor function is:
for arbitrary complex |

*q*| ≤ 1 and

*a*. This summation also appears as the Fourier series of the Eisenstein series and the invariants of the Weierstrass elliptic functions.
In little-o notation, the divisor function satisfies the inequality (see page 296 of Apostol’s book)
More precisely, Severin Wigert showed that
On the other hand, since there are infinitely many prime numbers,
In Big-O notation, Peter Gustav Lejeune Dirichlet showed that the average order of the divisor function satisfies the following inequality (see Theorem 3.3 of Apostol’s book)
where

is Euler's constant. Improving the bound

in this formula is known as Dirichlet's divisor problem
The behaviour of the sigma function is irregular. The asymptotic growth rate of the sigma function can be expressed by:
where lim sup is the limit superior. This result is

**Grönwall's theorem**, published in 1913 (Grönwall 1913). His proof uses Mertens' 3rd theorem, which says that
where

*p* denotes a prime.
In 1915, Ramanujan proved that under the assumption of the Riemann hypothesis, the inequality:
holds for all sufficiently large

*n* (Ramanujan 1997). In 1984 Guy Robin proved that the inequality is true for all

*n* ≥ 5,041 if and only if the Riemann hypothesis is true (Robin 1984). This is

**Robin's theorem** and the inequality became known after him. The largest known value that violates the inequality is

*n*=5,040. If the Riemann hypothesis is true, there are no greater exceptions. If the hypothesis is false, then Robin showed there are an infinite number of values of

*n* that violate the inequality, and it is known that the smallest such

*n* ≥ 5,041 must be superabundant (Akbary & Friggstad 2009). It has been shown that the inequality holds for large odd and square-free integers, and that the Riemann hypothesis is equivalent to the inequality just for

*n* divisible by the fifth power of a prime (Choie et al. 2007).
A related bound was given by Jeffrey Lagarias in 2002, who proved that the Riemann hypothesis is equivalent to the statement that
for every natural number

*n*, where

is the

*n*th harmonic number, (Lagarias 2002).
Robin also proved, unconditionally, that the inequality
holds for all

*n* ≥ 3.

**60** (**sixty**) () is the natural number following 59 and preceding 61. Being three times twenty, 60 is called "three score" in some older literature.
**Sixty** is a composite number with divisors 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60, making it also a highly composite number. Because 60 is the sum of its unitary divisors (excluding itself), it is a unitary perfect number, and it is also an excessive number with an abundance of 48. Being ten times a perfect number, 60 is a semiperfect number.
Sixty is the smallest number divisible by the numbers 1 to 6. (There is no smaller number divisible by the numbers 1 to 5). 60 is the smallest number with exactly 12 divisors. It is one of only 7 integers that have more divisors then any number twice itself (sequence in OEIS), and is one of only 6 that are also lowest common multiple of a consecutive set of integers from 1, and one of the 6 that are divisors of every highly composite number higher than itself.(sequence in OEIS)
Sixty is the sum of a pair of twin primes (29 + 31), as well as the sum of four consecutive primes (11 + 13 + 17 + 19). It is adjacent to two prime numbers (59,61). It is also the smallest number which is the sum of two odd primes in 6 ways.
The smallest non-solvable group (_{5}A) has order 60.
There are four Archimedean solids with 60 vertices: the truncated icosahedron, the rhombicosidodecahedron, the snub dodecahedron, and the truncated dodecahedron. The skeletons of these polyhedra form 60-node vertex-transitive graphs. There are also two Archimedean solids with 60 edges: the snub cube and the icosidodecahedron. The skeleton of the icosidodecahedron forms a 60-edge symmetric graph.
There are 60 one-sided hexominoes, the polyominoes made from 6 squares.
In geometry, 60 is the number of seconds in a minute, and the number of minutes in a degree. In normal space, the 3 interior angles of an equilateral triangle each measure 60 degrees, adding up to 180 degrees.
Because 60 is divisible by the sum of its digits in base 10, it is a Harshad number.
A number system with base 60 is called sexagesimal (the original meaning of *sexagesimal* is sixtieth).
The first fullerene to be discovered was buckminsterfullerene C_{60} – an allotrope of carbon with 60 atoms in each molecule, arranged in a truncated icosahedron. This ball is known as a buckyball, and looks like a soccer ball.
The atomic number of Neodymium is 60, and Cobalt-60 (60Co) is a radioactive isotope of cobalt.
The electrical utility frequency in western Japan, South Korea, Taiwan, the Philippines, Saudi Arabia, the United States, and several other countries in the Americas is 60 Hz.
An exbibyte (sometimes called exabyte) is 260 bytes.
The Babylonian number system had a base of sixty, inherited from the Sumerian and Akkadian civilizations, and possibly motivated by the large number of divisors which 60 has. The sexagesimal measurement of time and of geometric angles is a legacy of the Babylonian system.
The number system in the Mali Empire was also based on sixty (this is reflected in the counting system of the Maasina Fulfulde, a variant of the Fula language spoken in contemporary Mali). The Ekagi of Western New Guinea have also used base 60, and the sexagenary cycle also plays a role in Chinese calendar and numerology.
In German: and in Latin: refer to 60 = 5 dozen = 1/2 small gross. This quantity was used in international medieval treaties e.g. for ransom of captured Teutonic Knights.
In the Bible, the number 60 occurs several times, for example as the age of Isaac when Jacob and Esau were born, and the number of warriors escorting King Solomon.
In the laws of kashrut of Judaism, 60 is also the proportion (60:1) of kosher to non-kosher ingredients which can render an admixture kosher post-facto.
In the Koran, the number 60 is mentioned once: "..he should feed sixty indigent ones..", yet 60 is mentioned many times in the Hadith, most notably Muhammad being reported to say, "..Allah, the Exalted and Glorious, created Adam in His own image with His length of sixty cubits.."
**Sixty** is:

In mathematics, a

**divisor** of an integer

, also called a

**factor** of

, is an integer which divides

without leaving a remainder.
The name "divisor" comes from the arithmetic operation of division: if

then

is the dividend,

the

**divisor,** and

the quotient.
In general, for non-zero integers

and

, it is said that

**divides** —and, dually, that

is

**divisible** by

—written:
if there exists an integer

such that

. Thus, divisors can be negative as well as positive, although sometimes the term is restricted to positive divisors. (For example, there are six divisors of four, 1, 2, 4, −1, −2, −4, but only the positive ones would usually be mentioned, i.e. 1, 2, and 4.)
1 and −1 divide (are divisors of) every integer, every integer (and its negation) is a divisor of itself, and every integer is a divisor of 0, except by convention 0 itself (see also division by zero). Numbers divisible by 2 are called even and numbers not divisible by 2 are called odd.
1, −1,

*n* and −

*n* are known as the

**trivial divisors** of

*n*. A divisor of

*n* that is not a trivial divisor is known as a

**non-trivial divisor**. A number with at least one non-trivial divisor is known as a composite number, while the units −1 and 1 and prime numbers have no non-trivial divisors.
There are divisibility rules which allow one to recognize certain divisors of a number from the number's digits.
The generalization can be said to be the concept of

*divisibility* in any integral domain.
There are some elementary rules:
If

, and gcd

, then

. This is called Euclid's lemma.
If

is a prime number and

then

or

.
A positive divisor of

which is different from

is called a

**proper divisor** or an

**aliquot part** of

. A number that does not evenly divide

but leaves a remainder is called an

**aliquant part** of

.
An integer

whose only proper divisor is 1 is called a prime number. Equivalently, a prime number is a positive integer which has exactly two positive factors: 1 and itself.
Any positive divisor of

is a product of prime divisors of

raised to some power. This is a consequence of the fundamental theorem of arithmetic.
A number

is said to be perfect if it equals the sum of its proper divisors, deficient if the sum of its proper divisors is less than

, and abundant if this sum exceeds

.
The total number of positive divisors of

is a multiplicative function

, meaning that when two numbers

and

are relatively prime, then

. For instance,

; the eight divisors of 42 are 1, 2, 3, 6, 7, 14, 21 and 42. However the number of positive divisors is not a totally multiplicative function: if the two numbers

and

share a common divisor, then it might not be true that

. The sum of the positive divisors of

is another multiplicative function

(e.g.

). Both of these functions are examples of divisor functions.
If the prime factorization of

is given by
then the number of positive divisors of

is
and each of the divisors has the form
where

for each

For every natural

,

.
Also,
where

is Euler–Mascheroni constant. One interpretation of this result is that a randomly chosen positive integer

*n* has an expected number of divisors of about

.
The relation of divisibility turns the set

of non-negative integers into a partially ordered set, in fact into a complete distributive lattice. The largest element of this lattice is 0 and the smallest is 1. The meet operation ^ is given by the greatest common divisor and the join operation v by the least common multiple. This lattice is isomorphic to the dual of the lattice of subgroups of the infinite cyclic group

.

In number theory, a

**practical number** or

**panarithmic number** is a positive integer

*n* such that all smaller positive integers can be represented as sums of distinct divisors of

*n*. For example, 12 is a practical number because all the numbers from 1 to 11 can be expressed as sums of its divisors 1, 2, 3, 4, and 6: as well as these divisors themselves, we have 5=3+2, 7=6+1, 8=6+2, 9=6+3, 10=6+3+1, and 11=6+3+2.
The sequence of practical numbers (sequence in OEIS) begins
Practical numbers were used by Fibonacci in his Liber Abaci (1202) in connection with the problem of representing rational numbers as Egyptian fractions. Fibonacci does not formally define practical numbers, but he gives a table of Egyptian fraction expansions for fractions with practical denominators.
The name "practical number" is due to Srinivasan (1948), who first attempted a classification of these numbers that was completed by Stewart (1954) and Sierpiński (1955). This characterization makes it possible to determine whether a number is practical by examining its prime factorization. Any even perfect number and any power of two is also a practical number.
Practical numbers have also been shown to be analogous with prime numbers in many of their properties.
As Stewart (1954) and Sierpiński (1955) showed, it is straightforward to determine whether a number is practical from its prime factorization. A positive integer

with

and primes

is practical if and only if

and, for every

*i* from 2 to

*k*,
where

denotes the sum of the divisors of

*x*. For example, 3 ≤ σ(2)+1 = 4, 29 ≤ σ(2 × 32)+1 = 40, and 823 ≤ σ(2 × 32 × 29)+1=1171, so 2 × 32 × 29 × 823 = 429606 is practical. This characterization extends a partial classification of the practical numbers given by Srinivasan (1948).
It is not difficult to prove that this condition is necessary and sufficient for a number to be practical. In one direction, this condition is clearly necessary in order to be able to represent

as a sum of divisors of

*n*. In the other direction, the condition is sufficient, as can be shown by induction. More strongly, one can show that, if the factorization of

*n* satisfies the condition above, then any

can be represented as a sum of divisors of

*n*, by the following sequence of steps:
Any power of two is a practical number. Powers of two trivially satisfy the characterization of practical numbers in terms of their prime factorizations: the only prime in their factorizations,

*p*_{1}, equals two as required. Any even perfect number is also a practical number: due to Euler's result that these numbers must have the form 2 − 1

*n*(2

*n* − 1), every odd prime factor of an even perfect number must be at most the sum of the divisors of the even part of the number, and therefore the number must satisfy the characterization of practical numbers.
Any primorial is practical. By Bertrand's postulate, each successive prime in the prime factorization of a primorial must be smaller than the product of the first and last primes in the factorization of the preceding primorial, so primorials necessarily satisfy the characterization of practical numbers. Therefore, also, any number that is the product of nonzero powers of the first

*k* primes must also be practical; this includes Ramanujan's highly composite numbers (numbers with more divisors than any smaller positive integer) as well as the factorial numbers.
If

*n* is practical, then any rational number of the form

*m*/

*n* may be represented as a sum ∑

*d*_{i}/

*n* where each

*d*_{i} is a distinct divisor of

*n*. Each term in this sum simplifies to a unit fraction, so such a sum provides a representation of

*m*/

*n* as an Egyptian fraction. For instance,
Fibonacci, in his 1202 book

*Liber Abaci* lists several methods for finding Egyptian fraction representations of a rational number. Of these, the first is to test whether the number is itself already a unit fraction, but the second is to search for a representation of the numerator as a sum of divisors of the denominator, as described above; this method is only guaranteed to succeed for denominators that are practical. Fibonacci provides tables of these representations for fractions having as denominators the practical numbers 6, 8, 12, 20, 24, 60, and 100.
Vose (1985) showed that every number

*x*/

*y* has an Egyptian fraction representation with

terms. The proof involves finding a sequence of practical numbers

*n*_{i} with the property that every number less than

*n*_{i} may be written as a sum of

distinct divisors of

*n*_{i}. Then,

*i* is chosen so that

*n*_{i − 1} <

*y* ≤

*n*_{i}, and

*xn*_{i} is divided by

*y* giving quotient

*q* and remainder

*r*. It follows from these choices that

. Expanding both numerators on the right hand side of this formula into sums of divisors of

*n*_{i} results in the desired Egyptian fraction representation. Tenenbaum & Yokota (1990) use a similar technique involving a different sequence of practical numbers to show that every number

*x*/

*y* has an Egyptian fraction representation in which the largest denominator is

.
One reason for interest in practical numbers is that many of their properties are similar to properties of the prime numbers. For example, if

*p*(

*x*) is the enumerating function of practical numbers, i.e., the number of practical numbers not exceeding

*x*, Saias (1997) proved that for suitable constants

*c*_{1} and

*c*_{2}:
a formula which resembles the prime number theorem. This result largely resolved a conjecture of Margenstern (1991) that

*p*(

*x*) is asymptotic to

*cx*/log

*x* for some constant

*c*, and it strengthens an earlier claim of Erdős & Loxton (1979) that the practical numbers have density zero in the integers.
Theorems analogous to Goldbach's conjecture and the twin prime conjecture are also known for practical numbers: every positive even integer is the sum of two practical numbers, and there exist infinitely many triples of practical numbers

*x* − 2,

*x*,

*x* + 2. Melfi also showed that there are infinitely many practical Fibonacci numbers (sequence in OEIS); the analogous question of the existence of infinitely many Fibonacci primes is open. Hausman & Shapiro (1984) showed that there always exists a practical number in the interval [

*x*2,(

*x* + 1)2] for any positive real

*x*, a result analogous to Legendre's conjecture for primes.

In mathematics, the

**greatest common divisor** (

**gcd**), also known as the

**greatest common factor** (

**gcf**), or

**highest common factor** (

**hcf**), of two or more integers (at least one of which is not zero), is the largest positive integer that divides the numbers without a remainder. For example, the GCD of 8 and 12 is 4.
This notion can be extended to polynomials, see Polynomial greatest common divisor, or to rational numbers (with integer quotients).
In this article we will denote the greatest common divisor of two integers

*a* and

*b* as gcd(

*a*,

*b*). Some older textbooks use (

*a*,

*b*).
The number 54 can be expressed as a product of two other integers in several different ways:
Thus the

**divisors of 54** are:
Similarly

**the divisors of 24** are:
The numbers that these two lists share in common are the

**common divisors** of 54 and 24:
The greatest of these is 6. That is the

**greatest common divisor** of 54 and 24. One writes:
The greatest common divisor is useful for reducing fractions to be in lowest terms. For example, gcd(42, 56) = 14, therefore,
Two numbers are called

*relatively prime*, or

*coprime* if their greatest common divisor equals 1. For example, 9 and 28 are relatively prime.
For example, a 24-by-60 rectangular area can be divided into a grid of: 1-by-1 squares, 2-by-2 squares, 3-by-3 squares, 4-by-4 squares, 6-by-6 squares or 12-by-12 squares. Therefore, 12 is the greatest common divisor of 24 and 60. A 24-by-60 rectangular area can be divided into a grid of 12-by-12 squares, with two squares along one edge (24/12 = 2) and five squares along the other (60/12 = 5).
Greatest common divisors can in principle be computed by determining the prime factorizations of the two numbers and comparing factors, as in the following example: to compute gcd(18, 84), we find the prime factorizations 18 = 2 · 32 and 84 = 22 · 3 · 7 and notice that the "overlap" of the two expressions is 2 · 3; so gcd(18, 84) = 6. In practice, this method is only feasible for small numbers; computing prime factorizations in general takes far too long.
Here is another concrete example, illustrated by a Venn diagram. Suppose it is desired to find the greatest common divisor of 48 and 180. First, find the prime factorizations of the two numbers:
What they share in common is two "2"s and a "3":
A much more efficient method is the Euclidean algorithm, which uses a division algorithm such as long division in combination with the observation that the gcd of two numbers also divides their difference. To compute gcd(48,18), divide 48 by 18 to get a quotient of 2 and a remainder of 12. Then divide 18 by 12 to get a quotient of 1 and a remainder of 6. Then divide 12 by 6 to get a remainder of 0, which means that 6 is the gcd. Note that we ignored the quotient in each step except to notice when the remainder reached 0, signalling that we had arrived at the answer. Formally the algorithm can be described as:
where
If the arguments are both greater than zero then the algorithm can be written in more elementary terms as follows:
The existence of the Euclidean algorithm places (the decision problem version of) the greatest common divisor problem in P, the class of problems solvable in polynomial time. The GCD problem is not known to be in NC, and so there is no known way to parallelize its computation across many processors; nor is it known to be P-complete, which would imply that it is unlikely to be possible to parallelize GCD computation. In this sense the GCD problem is analogous to e.g. the integer factorization problem, which has no known polynomial-time algorithm, but is not known to be NP-complete. Shallcross et al. showed that a related problem (EUGCD, determining the remainder sequence arising during the Euclidean algorithm) is NC-equivalent to the problem of integer linear programming with two variables; if either problem is in

**NC** or is

**P-complete**, the other is as well. Since

**NC** contains NL, it is also unknown whether a space-efficient algorithm for computing the GCD exists, even for nondeterministic Turing machines.
Although the problem is not known to be in

**NC**, parallel algorithms with time superior to the Euclidean algorithm exist; the best known deterministic algorithm is by Chor and Goldreich, which (in the CRCW-PRAM model) can solve the problem in O(

*n*/log

*n*) time with

*n*1+ε processors. Randomized algorithms can solve the problem in O((log

*n*)2) time on

processors (note this is superpolynomial).
An alternative method of computing the gcd is the binary gcd method which uses only subtraction and division by 2. In outline the method is as follows: Let

*a* and

*b* be the two non negative integers and (without loss of generality) let

*a* ≥

*b*. There are now four possibilities
In this case 2 is a common factor. Divide both

*a* and

*b* by 2 and continue.
In this case 2 is not a common factor. Divide

*a* by 2 and continue.
Like the previous case 2 is not a common factor. Divide

*b* by 2 and continue.
In this case let c = ( a - b ) / 2. Then gcd(a,b) = gcd(a,c) = gcd(b,c). Because

*b* ≤

*a* it is usually easier (and computationally faster) to determine the gcd(b,c). If computing this algorithm by hand the gcd(b,c) may be apparent. Otherwise continue the algorithm until either

*b* =

*c* or

*c* = 0. For further details see Binary GCD algorithm.
If

*a* and

*b* are not both zero, the greatest common divisor of

*a* and

*b* can be computed by using least common multiple (lcm) of

*a* and

*b*:
but more commonly the lcm is computed from the gcd.
Using Thomae's function

*f*,
which generalizes to

*a* and

*b* rational or commensurate reals.
Keith Slavin has shown that for odd

*a* ≥ 1:
which is a function that can be evaluated for complex

*b*. Wolfgang Schramm has shown that
is an entire function in the variable

*b* for all positive integers

*a* where

*c*_{d}(

*k*) is Ramanujan's sum. Donald Knuth proved the following reduction:
for non-negative integers

*a* and

*b*, where

*a* and

*b* are not both zero. More generally
which can be proven by considering the Euclidean algorithm in base

*n*. Another useful identity relates

to the Euler's totient function:
In 1972, James E. Nymann showed that

*k* integers, chosen independently and uniformly from {

*1*,...,

*n*}, are coprime with probability 1/

*ζ*(

*k*) as

*n* goes to infinity. (See coprime for a derivation.) This result was extended in 1987 to show that the probability that

*k* random integers has greatest common divisor

*d* is

*d**-k*/ζ(

*k*).
Using this information, the expected value of the greatest common divisor function can be seen (informally) to not exist when

*k* = 2. In this case the probability that the gcd equals

*d* is

*d*−2/ζ(2), and since ζ(2) = π2/6 we have
This last summation is the harmonic series, which diverges. However, when

*k* ≥ 3, the expected value is well-defined, and by the above argument, it is
For

*k* = 3, this is approximately equal to 1.3684. For

*k* = 4, it is approximately 1.1106.
The notion of greatest common divisor can more generally be defined for elements of an arbitrary commutative ring, although in general there need not exist one for every pair of elements.
If

*R* is a commutative ring, and

*a* and

*b* are in

*R*, then an element

*d* of

*R* is called a

*common divisor* of

*a* and

*b* if it divides both

*a* and

*b* (that is, if there are elements

*x* and

*y* in

*R* such that

*d*·

*x* =

*a* and

*d*·

*y* =

*b*). If

*d* is a common divisor of

*a* and

*b*, and every common divisor of

*a* and

*b* divides

*d*, then

*d* is called a

*greatest common divisor* of

*a* and

*b*.
Note that with this definition, two elements

*a* and

*b* may very well have several greatest common divisors, or none at all. If

*R* is an integral domain then any two gcd's of

*a* and

*b* must be associate elements, since by definition either one must divide the other; indeed if a gcd exists, any one of its associates is a gcd as well. Existence of a gcd is not assured in arbitrary integral domains. However if

*R* is a unique factorization domain, then any two elements have a gcd, and more generally this is true in gcd domains. If

*R* is a Euclidean domain in which euclidean division is given algorithmically (as is the case for instance when

*R* =

*F*[

*X*] where

*F* is a field, or when

*R* is the ring of Gaussian integers), then greatest common divisors can be computed using a form of the Euclidean algorithm based on the division procedure.
The following is an example of an integral domain with two elements that do not have a gcd:
The elements 2 and 1 + √(−3) are two "maximal common divisors" (i.e. any common divisor which is a multiple of 2 is associated to 2, the same holds for 1 + √(−3)), but they are not associated, so there is no greatest common divisor of

*a* and

*b*.
Corresponding to the Bézout property we may, in any commutative ring, consider the collection of elements of the form

*pa* +

*qb*, where

*p* and

*q* range over the ring. This is the ideal generated by

*a* and

*b*, and is denoted simply (

*a*,

*b*). In a ring all of whose ideals are principal (a principal ideal domain or PID), this ideal will be identical with the set of multiples of some ring element

*d*; then this

*d* is a greatest common divisor of

*a* and

*b*. But the ideal (

*a*,

*b*) can be useful even when there is no greatest common divisor of

*a* and

*b*. (Indeed, Ernst Kummer used this ideal as a replacement for a gcd in his treatment of Fermat's Last Theorem, although he envisioned it as the set of multiples of some hypothetical, or

*ideal*, ring element

*d*, whence the ring-theoretic term.)

An **untouchable number** is a positive integer that cannot be expressed as the sum of all the proper divisors of any positive integer (including the untouchable number itself).
For example, the number 4 is not untouchable as it is equal to the sum of the proper divisors of 9: 1 + 3 = 4. The number 5 is untouchable as it is not the sum of the proper divisors of any positive integer: 5 = 1 + 4 is the only way to write 5 as the sum of distinct positive integers including 1, but if 4 divides a number, 2 does also, so 1 + 4 cannot be the sum of all of any number's proper divisors (since the list of factors would have to contain both 4 and 2).
The first few untouchable numbers are:
The number 5 is believed to be the only odd untouchable number, but this has not been proven: it would follow from a slightly stronger version of the Goldbach conjecture. Thus it appears that besides 2 and 5, all untouchable numbers are composite numbers. No perfect number is untouchable, since, at the very least, it can be expressed as the sum of its own proper divisors.
There are infinitely many untouchable numbers, a fact that was proven by Paul Erdős.
No untouchable number is one more than a prime number, since if *p* is prime, then the sum of the proper divisors of *p*2 is *p* + 1. Also, no untouchable number is three more than a prime number, except 5, since if *p* is prime (except two) then the sum of the proper divisors of *2p* is *p* + 3.

**28** (

**twenty-eight**) is the natural number following 27 and preceding 29.
There are also 28 days in February unless it is a leap year.
It is a composite number, its proper divisors being 1, 2, 4, 7, and 14.
Twenty-eight is the second perfect number. As a perfect number, it is related to the Mersenne prime 7, since 22(23 - 1) = 28. The next perfect number is 496, the previous being 6.
Twenty-eight is the third Granville number. The next Granville number is 96, the previous being 24.
Twenty-eight is the sum of the totient function for the first nine integers.
Twenty-eight is not the aliquot sum of any number other than itself and is therefore not a component in an aliquot sequence.
Since the greatest prime factor of 282 + 1 = 785 is 157, which is more than 28 twice, 28 is a Størmer number.
Twenty-eight is a harmonic divisor number, a happy number, a triangular number, a hexagonal number, and a centered nonagonal number.
It appears in the Padovan sequence, preceded by the terms 12, 16, 21 (it is the sum of the first two of these).
It is also a Keith number, because it recurs in a Fibonacci-like sequence started from its base 10 digits: 2, 8, 10, 18, 28...
Twenty-eight is the third positive integer with a prime factorization of the form

where

*q* is an odd prime.
Twenty-eight is the ninth and last number in early Indian magic square of order 3.
There are twenty-eight convex uniform honeycombs.
Twenty-eight is the only positive integer that has a unique Kayles nim-value.

**Twenty-eight** is: