Coprime integers

The numbers 8 and 9 are coprime, despite the fact that neither considered individually is a prime number, since 1 is their only common divisor. On the other hand, 6 and 9 are not coprime, because they are both divisible by 3. The numerator and denominator of a reduced fraction are coprime, by definition.

The numbers 1 and −1 are the only integers coprime with every integer, and they are the only integers that are coprime with 0.

The concept of pairwise coprimality is important as a hypothesis in many results in number theory, such as the Chinese remainder theorem.

The order of generation of coprime pairs by this algorithm. First node (2,1) is marked red, its three children are shown in orange, third generation is yellow, and so on in the rainbow order. There are coprime pairs near the axes and in some of the gaps but with dots too small to see here.

In machine design, an even, uniform gear wear is achieved by choosing the tooth counts of the two gears meshing together to be relatively prime. When a 1:1 gear ratio is desired, a gear relatively prime to the two equal-size gears may be inserted between them.