We find a formula for the probability that the product of n positive integers, chosen at random, is divisible by some integer d. We do this via an inductive application of the Chinese Remainder Theorem, generating functions, and several other combinatorial arguments. Additionally, we apply this formula to find a unique, but slow, probabilistic primality test.
Fine, Noah Y.
"Divisibility Probabilities for Products of Randomly Chosen Integers,"
Rose-Hulman Undergraduate Mathematics Journal: Vol. 24:
2, Article 10.
Available at: https://scholar.rose-hulman.edu/rhumj/vol24/iss2/10