Question: 2. Let p be a prime and let a E Fp - {0}. (a) Show that there exists a natural number m E N such

2. Let p be a prime and let a E Fp - {0}. (a) Show that there exists a natural number m E N such that am = 1. Hint: first show that the set {ak : k ( N} is finite.] (b) The order of a, denoted ord(a), is defined to be the smallest natural number n such that an = 1. What are the orders of the elements of Fp - {0} for p = 3, 5, 7. (c) Show that if am = 1 for some m E Z then n = ord(a) divides m. Hint: show that god(m, n) = n.] (d) Show that ord(a) divides p - 1. Hint: you may use Fermat's Little Theorem from Problem 11 in Tutorial Week 2.]

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