Mathematics 243, section 3 – Algebraic Structures Information on Exam 2 October 26, 2012

General Information

The second exam this semester will be given next Friday, November 2. The exam will cover the material we have discussed since the first exam, through class on Friday, October 26. This is the material from sections 1.7, 2.1, 2.2, 2.3, 2.4, 2.5, and 2.6. The topics to review are:

1) Relations, especially equivalence relations, and their properties

2) Properties of Z

3) Proof by mathematical induction

4) Divisibility. Know the statement of the Division Algorithm (Theorem 2.10 in the text) and its proof,

5) Prime numbers, prime factorizations and the Unique Factorization Theorem (Theorem 2.18),

6) Greatest common divisors and Euclid’s algorithm for gcd(a, b). Know the proof of Theorem 2.12 (existence of gcd(a, b) in S = {x ∈ Z : x = ma+nb, for some m, n ∈ Z}, and uniqueness of gcd).

7) Congruence mod n and the integers mod n: the set of congruence classes Z_{n} and the addition and multiplication mod n operations.

Some Review Problems

From Gilbert and Gilbert:

1) Section 1.7: 4, 8, 12

2) Section 2.2: 4, 6, 13, 14, 32, 33, 45

3) Section 2.3: problems like 1-16, proofs like 17-25, 34, 37

4) Section 2.4: problems like 2, 3, 16, 18, 20 (Note: In the text, (a, b) = gcd(a, b).)

5) Section 2.5: problems like 1, 3-24, 30, 34

6) Section 2.6: problems like 3,4,5,6,7

Review Session

I will be happy to run a review session before the exam. Tuesday October 31 or Thursday, November 1 are probably the best times for this for me, but I might also be able to do a session Wednesday evening.

Sample Exam Questions

Disclaimer: As always, the following questions indicate the range of topics that will be covered and the approximate level of difficulty of the exam questions. The actual exam questions may be organized differently, though.

I. Expect an induction proof like one of the following: A) Show using mathematical induction that

1^{2} + 3^{2} + 5^{2} + ⋯ + (2n − 1)^{2} = \frac{n(2n − 1)(2n + 1)}{3}

for all integers n ≥ 1. B) The basic triangle inequality is |x + y| ≤ |x| + |y|, a fact that follows for all x, y ∈ Z

from the basic order properties. Taking this as given show that for all n ≥ 2 and all x_{i} ∈ Z,

|x_{1} + x_{2} + ⋯ + x_{n}| ≤ |x_{1}| + |x_{2}| + ⋯ + |x_{n}|.

II. Let R be the relation on Z defined by xRy if (and only if 5|(x^{2} − y^{2}). A) Is R an equivalence relation? Prove your assertion. B) Show that xRy if and only if x ≡ y (mod 5) or x ≡ −y (mod 5). (Hint: Euclid’s

Lemma)

III. A) Given integers a, and b > 0, prove that there exist unique integers q and r such that

a = qb + r and 0 ≤ r < b. (You may apply the Well-Ordering Principle without justifying that.)

B) Find the quotient q and the remainder r as in part A for a = 4578 and b = 235.

IV. A) Give the definition of the greatest common divisor of the integers a, b. B) Prove that if S = {x ∈ Z : x = ma + nb, m, n ∈ Z}, then the smallest positive integer

in S is gcd(a, b). C) Find the integer d = gcd(488, 376) and express d in the form d = m · 488 + n · 76 for

integers m, n. D) Prove that if there is a solution of the congruence ax ≡ b (mod n) (where n > 1),

then gcd(a, n)|b.

V. A) Show that for all n ≥ 1, 9|(10^{n} − 1) and 11|(10^{n} − (−1)^{n}). (Hint: For the first

statement, show first that 10^{k+1} − 1 = 10^{k} · 9 + (10^{k} − 1).) B) Restate the result of part A as a congruence. C) (Extra Credit-type question) How can you test numbers for divisibility by 9 using the

base 10 digits? How could you test numbers for divisibility by 11 using the base 10 digits?

A) Find all integer solutions of the congruence 12x + 3 ≡ 7 mod 31. B) Find all solutions [x] of the equation [17][x] + [4] = [5] in Z_{29}. C) Show that [x] has a multiplicative inverse in Z_{n} (that is, a [y] such that [x][y] = [1] =

[y][x]) if and only if gcd(x, n) = 1. D) Which [x] ∈ Z_{18} have multiplicative inverses?