Coprime numbers, also known as relatively prime or mutually prime numbers, are two integers that have no common positive factor except 1. This means that any prime number that divides one of the integers does not divide the other integer, and vice versa. For example, 9 and 10 are coprime because their only common factor is 1. In contrast, 10 and 15 are not coprime because they have a common factor of 5.

Coprime numbers can also refer to a set of integers that share no common positive factor except 1. A stronger condition on a set of integers is pairwise coprime, which means that every pair of different integers in the set are coprime.

A fast way to determine whether two numbers are coprime is by using the Euclidean algorithm or its faster variants such as the binary GCD algorithm or Lehmers GCD algorithm.

The number of integers coprime with a positive integer n, between 1 and n, is given by Eulers totient function, also known as Eulers phi function, φ(n) .

Some facts about coprime numbers include:

• The numbers 1 and -1 are the only integers coprime with every integer, and they are the only integers that are coprime with 0.
• Any number paired with 1 forms a coprime pair.
• The least common multiple (LCM) of two coprime numbers is always equal to their product.
• Any two prime numbers are always coprime.
• Any two consecutive numbers are coprime.

In summary, coprime numbers are two integers that have no common positive factor except 1, and they can also refer to a set of integers that share no common positive factor except 1. Coprime numbers have several interesting properties and can be determined using the Euclidean algorithm or Eulers totient function.