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The Greatest Common Factor (GCF) of 58 and 12 is 2. The prime factors of 58 are 2 and 29, while 12 has 2 and 3 as its prime factors.

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In mathematics, **sexy primes** are prime numbers that differ from each other by six. For example, the numbers 5 and 11 are both sexy primes, because they differ by 6. If *p* + 2 or *p* + 4 (where p is the lower prime) is also prime, then the sexy prime is part of a prime triplet.
The term "sexy prime" stems from the Latin word for six: *sex*.
As used in this article, n# stands for the product 2 · 3 · 5 · 7 · … of all the primes ≤ n.
The sexy primes (sequences and in OEIS) below 500 are:
As of May 2009[update] the largest known sexy prime was found by Ken Davis and has 11,593 digits. The primes are (*p*, *p*+6) for
9001# = 2×3×5×...×9001 is a primorial, i.e., the product of primes ≤ 9001.
Sexy primes can be extended to larger constellations. Triplets of primes (*p*, *p* + 6, *p* + 12) such that *p* + 18 is composite are called **sexy prime triplets**. Those below 1000 are (, , ):
As of April 2006[update] the largest known sexy prime triplet, found by Ken Davis had 5132 digits:
Sexy prime quadruplets (*p*, *p* + 6, *p* + 12, *p* + 18) can only begin with primes ending in a 1 in their decimal representation (except for the quadruplet with *p* = 5). The sexy prime quadruplets below 1000 are (, , , ):
In November 2005 the largest known sexy prime quadruplet, found by Jens Kruse Andersen had 1002 digits:
In September 2010 Ken Davis announced a 1004-digit quadruplet with *p* = 23333 + 1582534968299.
In an arithmetic progression of five terms with common difference 6, because 6>5 and the two numbers are relatively prime, one of the terms must be divisible by 5. Thus, the only sexy prime quintuplet is (5,11,17,23,29) with no longer sequence of sexy primes possible.

In number theory, the **prime factors** of a positive integer are the prime numbers that divide that integer exactly. The prime factorization of a positive integer is a list of the integer's prime factors, together with their multiplicities; the process of determining these factors can also be referred to as "factorization". The fundamental theorem of arithmetic says that every positive integer has a single unique prime factorization.
To shorten prime factorizations, factors are often expressed in powers (multiplicities). For example,
in which the factors 2, 3 and 5 have multiplicities of 3, 2 and 1, respectively.
For a prime factor *p* of *n*, the multiplicity of *p* is the largest exponent *a* for which *pa* divides *n* exactly.
For a positive integer *n*, the *number* of prime factors of *n* and the *sum* of the prime factors of *n* (not counting multiplicity) are examples of arithmetic functions of *n* that are additive but not completely additive.
Perfect square numbers can be recognized by the fact that all of their prime factors have even multiplicities. For example, the number 144 (the square of 12) has the prime factors
These can be rearranged to make the pattern more visible:
Because every prime factor appears an even number of times, the original number can be expressed as the square of some smaller number. In the same way, perfect cube numbers will have prime factors whose multiplicities are multiples of three, and so on.
Positive integers with no prime factors in common are said to be coprime. Two integers *a* and *b* can also be defined as coprime if their greatest common divisor gcd(*a*, *b*) = 1. Euclid's algorithm can be used to determine whether two integers are coprime without knowing their prime factors; the algorithm runs in a time that is polynomial in the number of digits involved.
The integer 1 is coprime to every positive integer, including itself. This is because it has no prime factors; it is the empty product. This implies that gcd(1, *b*) = 1 for any *b* ≥ 1.
Determining the prime factors of a number is an example of a problem frequently used to ensure cryptographic security in encryption systems; this problem is believed to require super-polynomial time in the number of digits — it is relatively easy to construct a problem that would take longer than the known age of the universe to solve on current computers using current algorithms.
A prime factor can be visualized by understanding Euclid's geometric position. He saw a whole number as a line segment, which has a smallest line segment greater than 1 that can divide equally into it.
The function *("omega")* represents the number of *distinct* prime factors of *n*, while the function *("big omega")* represents the *total* number of prime factors of . If
then
For example, , so and .
for = 1, 2, 3, ... is 0, 1, 1, 1, 1, 2, 1, 1, 1, ... (sequence in OEIS).
for = 1, 2, 3, ... is 0, 1, 1, 2, 1, 2, 1, 3, 2, ... (sequence in OEIS).

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

.

**Greatest common divisor**
In number theory, an

**aurifeuillian factorization** is a factorization of the form
Aurifeuille discovered the factorization for

*n* = 14 in 1871, as the following:
The second factor is prime, and the factorization of the first factor is

The general form of the factorization was later discovered by Lucas.

**58** (**fifty-eight**) is the natural number following 57 and preceding 59.
**Fifty-eight** has an aliquot sum of 32 and is the second composite member of the 31-aliquot tree.
**Fifty-eight** is the sum of the first seven prime numbers, an 11-gonal number, and a Smith number. Given 58, the Mertens function returns 0.
There is no solution to the equation *x* - φ(*x*) = 58, making 58 a noncototient. However, the sum of the totient function for the first thirteen integers is 58.
In the NBA, the most points ever scored in a fourth quarter was 58 by the Buffalo Braves (at Boston Celtics), Oct. 20, 1972. The Most points in a game by a rookie player: Wilt Chamberlain, 58: Philadelphia vs. Detroit, Jan. 25, 1960, and Philadelphia. vs. New York Knicks, Feb. 21, 1960
In MotoGP, 58 was the number of Marco Simoncelli who died in a fatal accident at the Malaysian Round of the 2011 MotoGP season. MotoGP's governing body, the FIM, are considering to retire number 58 from use in MotoGP as they did before with the numbers 74 and 48 of Daijiro Kato and Shoya Tomizawa respectively.
The number 58 was commonly associated with misfortune in many civilizations native to either Central America or Southern America. Due to their beliefs in the original 58 sins, the number came to symbolize curses and ill-luck. Aztec oracles supposedly stumbled across the number an unnaturally high number of times before disaster fell. One famous recording of this, though largely discredited as mere folktale, concerned the oracle of Moctezuma II, who allegedly counted 58 pieces of gold scattered before a sacrificial pit the day prior to the arrival of Hernan Cortés.

An **extravagant number** (also known as a *wasteful* number) is a natural number that has fewer digits than the number of digits in its prime factorization (including exponents). For example, in base-10 arithmetic 4 = 2², 6 = 2×3, 8 = 2³, and 9 = 3² are extravagant numbers.
Extravagant numbers can be defined in any base. There are infinitely many extravagant numbers, no matter what base is used.

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