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Probability of Coprimality

This one is rather easy to pose, and can be difficult to solve, even with a proof sitting right in front of you. The result is a bit surprising as well, and gives some insight into the structure of the integers and prime numbers.

Given two random integers, what is the probability that they have no common factors besides 1?

or, how it is usually posed:

What is the probability of two random integers being coprime?

Where “coprime” means “the greatest common divisor is one”.

coprime number distribution