Two integers are relatively prime if they share no common positive factors (divisors) except 1. Using the notation to denote the greatest
common divisor, two integers and are relatively prime if . Relatively prime integers are sometimes also called
strangers or coprime and are denoted . The plot above plots and along the two axes and colors a square black if and white otherwise (left figure) and simply colored
according to
(right figure).

Two numbers can be tested to see if they are relatively prime in the Wolfram Language using CoprimeQ[m,
n].

Two distinct primes
and
are always relatively prime, , as are any positive integer powers of distinct primes
and , .

Relative primality is not transitive. For example, and , but .

The probability that two integers and picked at random are relatively prime is

(1)

(OEIS A059956; Cesàro and Sylvester 1883; Lehmer 1900; Sylvester 1909; Nymann 1972; Wells 1986, p. 28; Borwein and
Bailey 2003, p. 139; Havil 2003, pp. 40 and 65; Moree 2005), where is the Riemann
zeta function. This result is related to the fact that the greatest
common divisor of
and ,
, can be interpreted as the number
of lattice points in the plane
which lie on the straight line connecting the vectors and (excluding itself). In fact, is the fractional number of lattice
pointsvisible from the origin
(Castellanos 1988, pp. 155-156).

Given three integers chosen at random, the probability that no common factor
will divide them all is

(2)

(OEIS A088453; Wells 1986, p. 29), where is Apéry's
constant (Wells 1986, p. 29). In general, the probability that random numbers lack a th power common divisor is (Cohen 1959, Salamin 1972, Nymann 1975, Schoenfeld
1976, Porubský 1981, Chidambaraswamy and Sitaramachandra Rao 1987, Hafner
et al. 1993).

Interestingly, the probability that two Gaussian integers
and
are relatively prime is

Amazingly, the probabilities for random pairs of integers and Gaussian integers being relatively prime are the same as the asymptotic densities of squarefree
integers of these types.

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