2.8 Find rules in the spirit of Proposition 2.1.9 for divisibility of an integer by 5, 9, and 11, and prove each of these rules using arithmetic modulo a suitable n.

Proficiency #7. [Counterexamples] For each of the following two statements, give a counterexample to show that the state- ment is false. a. If x is an integer such that x? = 1 mod 12, then x =1 mod 12 or x = -1 mod 12. b. If A, B, D are sets such that A C B and B Z D, then A Z D.

. (a) If g is a primitive root modulo 37, which of the numbers g², g³,..., g8 are primitive roots modulo 37?

For any integer n, either n2 mod 4 = 0, or n² mod 4 = 1.