Question

1. Let p be any prime number. Let r be any integer such that 0 < r < p−1. Show that there exists a number q such that rq = 1(mod p)

2. Let p1 and p2 be two distinct prime numbers. Let r1 and r2 be such that 0 < r1 < p1 and 0 < r2 < p2. Show that there exists a number x such that x = r1(mod p1)andx = r2(mod p2).

8. Suppose we roll a standard six-sided dice over and over again. What is the expected number of rolls until we got the first pair of consecutive sixs?

Answer 1