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Encyclopedia > Coprime

In mathematics, the integers a and b are said to be coprime or relatively prime if they have no common factor other than 1 and −1, or equivalently, if their greatest common divisor is 1. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... The integers are commonly denoted by the above symbol. ... In mathematics, a divisor of an integer n, also called a factor of n, is an integer which evenly divides n without leaving a remainder. ... In mathematics, the greatest common divisor (gcd), sometimes known as the greatest common factor (gcf) or highest common factor (hcf), of two non-zero integers, is the largest positive integer that divides both numbers without remainder. ...


For example, 6 and 35 are coprime, but 6 and 27 are not because they are both divisible by 3. The number 1 is coprime to every integer. Look up six in Wiktionary, the free dictionary. ... 35 (thirty-five) is the natural number following 34 and preceding 36. ... 27 (twenty-seven) is the natural number following 26 and preceding 28. ... This article is about the number one. ...


A fast way to determine whether two numbers are coprime is given by the Euclidean algorithm. In number theory, the Euclidean algorithm (also called Euclids algorithm) is an algorithm to determine the greatest common divisor (GCD) of two elements of any Euclidean domain (for example, the integers). ...


Euler's totient function (or Euler's phi function) of a positive integer n is the number of integers between 1 and n which are coprime to n. The first thousand values of φ(n) In number theory, the totient (n) of a positive integer n is defined to be the number of positive integers less than or equal to n and coprime to n. ...

Contents

Properties

Figure 1. The numbers 4 and 9 are coprime because the diagonal does not intersect any other lattice points

There are a number of conditions which are equivalent to a and b being coprime: Image File history File links Coprime-lattice. ... Image File history File links Coprime-lattice. ...

As a consequence, if a and b are coprime and brbs (mod a), then rs (mod a) (because we may "divide by b" when working modulo a). Furthermore, if a and b1 are coprime, and a and b2 are coprime, then a and b1b2 are also coprime (because the product of units is a unit). In number theory, Bézouts identity, named after Étienne Bézout, is a linear diophantine equation. ... The reciprocal function: y = 1/x. ... Modular arithmetic (sometimes called modulo arithmetic, or clock arithmetic because of its use in the 24-hour clock system) is a system of arithmetic for integers, where numbers wrap around after they reach a certain value — the modulus. ... In mathematics, a unit in a ring R is an element u such that there is v in R with uv = vu = 1R. That is, u is an invertible element of the multiplicative monoid of R. The units of R form a group U(R) under multiplication, the group of... In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ... Modular arithmetic (sometimes called modulo arithmetic, or clock arithmetic because of its use in the 24-hour clock system) is a system of arithmetic for integers, where numbers wrap around after they reach a certain value — the modulus. ... Modular arithmetic (sometimes called modulo arithmetic, or clock arithmetic because of its use in the 24-hour clock system) is a system of arithmetic for integers, where numbers wrap around after they reach a certain value — the modulus. ...


If a and b are coprime and a divides a product bc, then a divides c. This can be viewed as a generalisation of Euclid's lemma, which states that if p is prime, and p divides a product bc, then either p divides b or p divides c. Euclids lemma is a generalisation of Proposition 30 of Book VII of Euclids Elements. ...


The two integers a and b are coprime if and only if the point with coordinates (a, b) in a Cartesian coordinate system is "visible" from the origin (0,0), in the sense that there is no point with integer coordinates between the origin and (a, b). (See figure 1.) Fig. ...


The probability that two randomly chosen integers are coprime is 6/π2 (see pi), which is about 60%. See below. Probability is the likelihood that something is the case or will happen. ... When a circles diameter is 1, its circumference is Ï€. Pi or Ï€ is the ratio of a circles circumference to its diameter in Euclidean geometry, approximately 3. ...


Two natural numbers a and b are coprime if and only if the numbers 2a − 1 and 2b − 1 are coprime. In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ...


Cross notation, group

If n≥1 is an integer, the numbers coprime to n, taken modulo n, form a group with multiplication as operation; it is written as (Z/nZ)× or Zn*. The integers are commonly denoted by the above symbol. ... Modular arithmetic (sometimes called modulo arithmetic, or clock arithmetic because of its use in the 24-hour clock system) is a system of arithmetic for integers, where numbers wrap around after they reach a certain value — the modulus. ... This picture illustrates how the hours on a clock form a group under modular addition. ...


Generalizations

Two ideals A and B in the commutative ring R are called coprime if A + B = R. This generalizes Bézout's identity: with this definition, two principal ideals (a) and (b) in the ring of integers Z are coprime if and only if a and b are coprime. In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring which generalizes important properties of integers. ... In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ... In ring theory, a branch of abstract algebra, a ring is an algebraic structure in which addition and multiplication are defined and have similar properties to those familiar from the integers. ... In number theory, Bézouts identity, named after Étienne Bézout, is a linear diophantine equation. ... In Ring theory, a branch of abstract algebra, a principal ideal is an ideal I in a ring R that is generated by a single element a of R. More specifically: a left principal ideal of R is a subset of R of the form Ra := {ra : r in R...


If the ideals A and B of R are coprime, then AB = AB; furthermore, if C is a third ideal such that A contains BC, then A contains C. The Chinese remainder theorem is an important statement about coprime ideals. Several related results in number theory and abstract algebra are known under the name Chinese remainder theorem. ...


The concept of being relatively prime can also be extended any finite set of integers S = {a1, a2, .... an} to mean that the greatest common divisor of the elements of the set is 1. If every pair of integers in the set is relatively prime, then the set is called pairwise relatively prime. In mathematics, a set is called finite if there is a bijection between the set and some set of the form {1, 2, ..., n} where is a natural number. ... In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ... In mathematics, the greatest common divisor (gcd), sometimes known as the greatest common factor (gcf) or highest common factor (hcf), of two non-zero integers, is the largest positive integer that divides both numbers without remainder. ... In mathematics, especially number theory, a set of integers is said to be pairwise coprime (or pairwise relatively prime) if every pair of integers a and b in the set are coprime (that is, have no common divisors other than 1). ...


Every pairwise relatively prime set is relatively prime; however, the converse is not true: {6, 10, 15} is relatively prime, but not pairwise relative prime. (In fact, each pair of integers in the set has a non-trivial common factor.)


Probabilities

Given two randomly chosen integers A and B, it is reasonable to ask how likely it is that A and B are coprime. In this determination, it is convenient to use the characterization that A and B are coprime if and only if no prime number divides both of them (see Fundamental theorem of arithmetic). In number theory, the fundamental theorem of arithmetic (or unique factorization theorem) states that every natural number either is itself a prime number, or can be written as a unique product of prime numbers. ...


The probability that any number is divisible by a prime (or any integer), p is 1 / p. Hence the probability that two numbers are both divisible by this prime is 1 / p2, and the probability that at least one of them is not is 1 − 1 / p2. Thus the probability that two numbers are coprime is given by a product over all primes,

prod_p^{infty} left(1-frac{1}{p^2}right) = left( prod_p^{infty} frac{1}{1-p^{-2}} right)^{-1} = frac{1}{zeta(2)} = frac{6}{pi^2}.

Here ζ refers to the Riemann zeta function. In general, the probability of k randomly chosen integers being coprime is 1 / ζ(k). In mathematics, the Riemann zeta function, named after German mathematician Bernhard Riemann, is a function of significant importance in number theory, because of its relation to the distribution of prime numbers. ...


There is often confusion about what a "randomly chosen integer" is. One way of understanding this is to assume that the integers are chosen randomly between 1 and an integer N. Then for each upper bound N, there is a probability PN that two randomly chosen numbers are coprime. This will never be exactly 6 / π2, but in the limit as N to infty, P_N to 6/pi^2.


See also


  Results from FactBites:
 
Coprime - Wikipedia, the free encyclopedia (550 words)
In mathematics, the integers a and b are said to be coprime or relatively prime if they have no common factor other than 1 and −1, or equivalently, if their greatest common divisor is 1.
For example, 6 and 35 are coprime, but 6 and 27 are not because they are both divisible by 3.
The two integers a and b are coprime if and only if the point with coordinates (a, b) in an Cartesian coordinate system is "visible" from the origin (0,0), in the sense that there is no point with integer coordinates between the origin and (a, b).
  More results at FactBites »

 
 

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