For a prime , two integers are both divisible by with the probability , because this only happens when the two integers have the residue 0 (one out of available residues) modulo .
Two integers are mutually prime if they have no common nontrivial factors, prime facors in particular. Assuming divisibility by one prime is independent of divisibility by another, two integers are mutually prime with the probability
where the product is over all prime .
(There is an extended version of this argument - in Spanish.)
- TOM M. APOSTOL, What Is the Most Surprising Result in Mathematics? Part II, Math Horizons, Vol. 4, No. 3 (February 1997), pp. 26-31