Coprimality

Coprimality, or relative primality, is a relation between integers. Two integers a and b are coprime if their greatest common divisor gcd(a,b) equals 1. Equivalently, they have no common positive divisor greater than 1. The notion extends to negative integers by using absolute values: gcd(a,b)=gcd(|a|,|b|). If either number is zero, gcd(a,0)=|a|, so a is coprime with 0 only when |a|=1.

Equivalent characterizations include that a and b have disjoint prime factorizations; no prime divides both; and every common divisor divides 1. The concept can be generalized to more than two numbers: a set is pairwise coprime if every pair is coprime, and collectively coprime if the gcd of all numbers is 1.

Important consequences include Bezout's identity: if gcd(a,b)=1, there exist integers x and y with ax+by=1. This implies that a has a multiplicative inverse modulo b (and vice versa) whenever gcd(a,b)=1. For nonzero a and b, coprimality is symmetric: gcd(a,b)=gcd(b,a).

Examples help illustrate: gcd(6,35)=1, so 6 and 35 are coprime; gcd(6,15)=3, not coprime. With zero: gcd(1,0)=1, so 1 is coprime with 0; gcd(0,0) is undefined or 0, not coprime.

Several related ideas appear in number theory. The probability that two randomly chosen integers are coprime is 6/π^2. Euler’s totient function φ(n) counts integers up to n that are coprime to n. If a and b are coprime and c is any integer, gcd(ac,b)=1, and gcd(a,b,c)=1 expresses collective coprimality among multiple numbers.