§2.2 Exercises

Exercises 2.2.A

Write out the addition and multiplication tables for:

Solve the equation \(x^2 \oplus x = [0]\) in \(\mathbb{Z}_4\).

Solve the equation \(x^2 = [1]\) in \(\mathbb{Z}_8\).

Solve the equation \(x^4 = [1]\) in \(\mathbb{Z}_5\).

Solve the equation \(x^2 \oplus [3] \odot x \oplus [2] = [0]\) in \(\mathbb{Z}_6\).

Solve the equation \(x^2 \oplus [8] \odot x = [0]\) in \(\mathbb{Z}_9\).

Solve the equation \(x^3 \oplus x^2 \oplus x \oplus [1] = [0]\) in \(\mathbb{Z}_8\).

Solve the equation \(x^3 + x^2 = [2]\) in \(\mathbb{Z}_{10}\).

(a) Find an element \([a]\) in \(\mathbb{Z}_7\) such that every nonzero element of \(\mathbb{Z}_7\) is a power of \([a]\).

(b) Do part (a) in \(\mathbb{Z}_5\).

Prove parts 3, 7, 8, and 9 of Theorem 2.7.

Solve the following equations:

Prove or disprove: If \([a] \odot [b] = [0]\) in \(\mathbb{Z}_n\), then \([a] = [0]\) or \([b] = [0]\).

Prove or disprove: If \([a] \odot [b] = [a] \odot [c]\) and \([a] \neq [0]\) in \(\mathbb{Z}_n\), then \([b] = [c]\).

Solve the following equations:

(a) \(x^2 + x = [0]\) in \(\mathbb{Z}_5\)
(b) \(x^2 + x = [0]\) in \(\mathbb{Z}_6\)
(c) If \(p\) is prime, prove that the only solutions of \(x^2 + x = [0]\) in \(\mathbb{Z}_p\) are \([0]\) and \([p - 1]\).

Compute the following products:

(a) \(([a] \oplus [b])^2\) in \(\mathbb{Z}_2\)
(b) \(([a] \oplus [b])^3\) in \(\mathbb{Z}_3\) [Hint: Exercise 11(a) may be helpful.]
(c) \(([a] \oplus [b])^5\) in \(\mathbb{Z}_5\) [Hint: See Exercise 11(c).]
(d) Based on the results of parts (a)–(c), what do you think \(([a] \oplus [b])^r\) is equal to in \(\mathbb{Z}_r\)?

(a) Find all \([a]\) in \(\mathbb{Z}_5\) for which the equation \([a] \odot x = [1]\) has a solution. Then do the same thing for: