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.
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.
A fast way to determine whether two numbers are coprime is given by the Euclidean algorithm.
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''.

Contents

Properties

Cross notation, group

Generalizations

Probabilities

See also

Properties

There are a number of conditions which are equivalent to ''a'' and ''b'' being coprime:

★ There exist integers ''x'' and ''y'' such that ''ax'' + ''by'' = 1 (see Bézout's identity).

★ The integer ''b'' has a multiplicative inverse modulo ''a'': there exists an integer ''y'' such that ''by'' ≡ 1 (mod ''a''). In other words, ''b'' is a unit in the ring 'Z'/''a'''Z' of integers modulo ''a''.
As a consequence, if ''a'' and ''b'' are coprime and ''br'' ≡ ''bs'' (mod ''a''), then ''r'' ≡ ''s'' (mod ''a'') (because we may "divide by ''b''" when working modulo ''a''). Furthermore, if ''a'' and ''b''₁ are coprime, and ''a'' and ''b''₂ are coprime, then ''a'' and ''b''₁''b''₂ are also coprime (because the product of units is a unit).
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''.
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.)
The probability that two randomly chosen integers are coprime is 6/π² (see pi), which is about 60%. See below.
Two natural numbers ''a'' and ''b'' are coprime if and only if the numbers 2^''a'' − 1 and 2^''b'' − 1 are coprime.

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'/''n'''Z')^× or 'Z'_n
^★ .

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.
If the ideals ''A'' and ''B'' of ''R'' are coprime, then ''AB'' = ''A''∩''B''; 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.
The concept of being ''relatively prime'' can also be extended any finite set of integers ''S'' = {''a''₁, ''a''₂, .... ''a''_''n''} 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''.
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).
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 $1/p^2$, and the probability that at least one of them is not is $1-1/p^2$. Thus the probability that two numbers are coprime is given by a product over all primes,
:
ight) = left( prod_p^{infty} rac{1}{1-p^{-2}}
ight)^{-1} = rac{1}{zeta(2)} = rac{6}{pi^2}.
Here $zeta$ refers to the Riemann zeta function. In general, the probability of $k$ randomly chosen integers being coprime is $1/zeta(k)$.
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 $P\_N$ that two randomly chosen numbers are coprime. This will never be exactly $6/pi^2$, but in the limit as $N\; o\; infty$, $P\_N\; o\; 6/pi^2$.

See also

★ Greatest common divisor

★ Prime number