83. What is the probability that a randomly chosen positive integer will be coprime with 6? What is the probability that at least one of two randomly chosen numbers will be coprime with 6?
83. Among any six consecutive integers, one is divisible by 6, and the others give remainders of 1, 2, 3, 4, and 5 when divided by 6. Thus, among any six consecutive integers, two are coprime with 6 (namely, those that give remainders of 1 and 5 when divided by 6). Therefore, the probability that a randomly chosen number will be coprime with 6 is $\frac{2}{6}=\frac{1}{3}$.

From the fact that among any three numbers, one is coprime with 6, it is not difficult to derive that the probability that at least one of two randomly chosen numbers will be coprime with 6 is $\frac{5}{9}$ (cf. the solution to problem 73))).