Problem

(a) An integer N is to be selected at random from {1,2,..., (10)3} in the sense that each...

(a) An integer N is to be selected at random from {1,2,..., (10)3} in the sense that each integer has the same probability of being selected. What is the probability that N will be divisible by 3? by 5? by 7? by 15? by 105? How would your answer change if (10)3 is replaced by (10)k as k became larger and larger?

(b) An important function in number theory−one whose properties can be shown to be related to what is probably the most important unsolved problem of mathematics, the Riemann hypothesis−is the Möbius function μ(n), defined for all positive integral values n as follows: Factor n into its prime factors. Tf there is a repeated prime factor, as in 12 = 2 • 2 • 3 or 49 = 7 • 7, then μ(n) is defined to equal 0. Now let N be chosen at random from {1,2,... (10)k}, where k is large. Determine P{μ(n) = 0} as k →∞.