What exactly do you mean by "find a general rule" in Problem 3.3.3?
Look at the case with m=3. 20 = 1, 21 = 2, 22 = 1, and the sequence continues 1, 2, 1, 2, 1, 2,..., suggesting the general rule "2i = 1 for even i, 2i = 2 for odd i". You should be able to find similar rules for the other values of m. A hint: The rules of exponents, such as 2a+b = 2a2b, should be helpful in justifying your rules.
I'm not sure what's being asked in 3.5.3 (a). Isn't this just the same as the Chinese Remainder Theorem?
The CRT says that if you have a system of congruence where the bases are pairwise relatively prime, the system is equivalent to a single congruence. Here in 3.5.3 (a) you are given a single congruence and asked to show that there is a system of congruences, with prime-power bases, that is equivalent. Once you describe the system, you may quote to CRT to show that it is equivalent to the original congruence.
The definition at the start of Problem 3.5.3 doesn't seem to be referred to later, is this a mistake?
It's not referred to in (a) or (b), but it defines what is meant in (c) by "an arbitrary system of congruences".
Last modified 4 March 2005