# CMPSCI 250: Introduction to Computation

### David Mix Barrington

### Spring, 2005

# Homework #3 Questions and Answers

#### Question 3.2, 4 March 2005

What exactly do you mean by "find a general rule" in Problem 3.3.3?

Look at the case with m=3. 2^{0} = 1,
2^{1} = 2, 2^{2} = 1, and the sequence continues 1, 2, 1, 2,
1, 2,..., suggesting the general rule "2^{i} = 1 for even i,
2^{i} = 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
2^{a+b} = 2^{a}2^{b}, should be helpful in justifying
your rules.

#### Question 3.1, 2 March 2005

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