Question: 1. Let n N. Explain why the additive order of any x Z/nZ is a divisor of n, and prove that for any d

1. Let n N. Explain why the additive order of any x ZZ is a divisor of n, and prove that for any d | n,

1. Let n N. Explain why the additive order of any x Z/nZ is a divisor of n, and prove that for any d | n, there exists an x E Z/nZ of order d. 2. Let p E N be a prime. Explain why the multiplicative order of any x E (Z/pZ) is a divisor of p - 1, and prove that for any d | (p-1), there exists an x E (Z/pZ)x of multiplicative order d. 3. Let n N. Is it true that for any d | (n), there exists an x (Z/nZ) of multiplicative order d? 4. Suppose that n N, and that there exists an x (Z/nZ) of multiplicative order n 1. Prove that n must be prime.

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