## Coprime numbers (also called: numbers prime to each other, relatively prime, mutually prime)

- The number "a" and "b" are said to be
**relatively prime, mutually prime, or coprime**if the only positive integer that evenly divides both of them is 1. - The coprime numbers are pairs of (at least two) numbers that do not have any other common factor than 1.
- When the only common factor is 1, then this is also equivalent to their greatest (highest) common factor (divisor) being 1.

**Examples of pairs of coprime numbers:**- The coprime numbers are not necessarily prime numbers themselves, for example 4 and 9 - these two numbers are not prime, they are composite numbers, since 4 = 2 × 2 = 2
^{2}and 9 = 3 × 3 = 3^{2}. But the gcf (4, 9) = 1, so they are coprime, or prime to each other, or relatively prime. - Sometimes, the coprime numbers in a pair are prime numbers themselves, for example (3 and 5), or (7 and 11), (13 and 23).
- Some other times, the numbers that are prime to each other may or may not be prime, for example (5 and 6), (7 and 12), (15 and 23).

**Examples of pairs of numbers that are not coprime:**- 16 and 24 are not coprime, since they are both divisible by 1, 2, 4 and 8 (1, 2, 4 and 8 are their common factors).
- 6 and 10 are not coprime, since they are both divisible by 1 and 2.

**Some properties of the coprime numbers:**- The greatest (highest) common factor (divisor) of two coprime numbers is always 1.
- The least common multiple, LCM, of two coprimes is always their product: LCM (a, b) = a × b.
- The numbers 1 and -1 are the only integers coprime to every integer, for example (1 and 2), (1 and 3), (1 and 4), (1 and 5), (1 and 6), and so on, are pairs of coprime numbers since their greatest (highest) common factor (divisor) is 1.
- The numbers 1 and -1 are the only integers coprime to 0.
- Any two prime numbers are always coprime, for example (2 and 3), (3 and 5), (5 and 7) and so on.
- Any two consecutive numbers are co-prime, for example (1 and 2), (2 and 3), (3 and 4), (4 and 5), (5 and 6), (6 and 7), (7 and 8), (8 and 9), (9 and 10), and so on.
- The sum of two coprime numbers, a + b, is always relatively prime to their product, a × b. For example, 7 and 10 are coprime numbers, 7 + 10 = 17 is relatively prime to 7 × 10 = 70. Another example, 9 and 11 are coprime, and their sum, 9 + 11 = 20 is relatively prime to their product, 9 × 11 = 99.
- A quick way to determine whether two numbers are prime to each other is given by the Euclidean algorithm: The Euclidean Algorithm