8.1 If we take the linear congruential algorithm with an additive component of 0, Xn+1 = (aXn) mod m Then it can be shown that if m is prime and if a given value of a produces the maximum period of m - 1, then ak will also produce the maximum period, provided that k is less than m and that k and m - 1 are relatively prime. Demonstrate this by using X0 = 1 and m = 31 and producing the sequences for ak = 3, 32, 33, and 34.

a. What is the maximum period obtainable from the following generator? Xn+1 = (aXn) mod 24

b. What should be the value of a?

c. What restrictions are required on the seed?