Mathematics 243, section 2 – Mathematical Structures Information on Exam 2 October 25, 2017

General Information

The second exam this semester will be given next week, either on Thursday evening, November 2 or in class on Friday morning, November 3. (We’ll discuss the options on October 27 and decide.) The exam will cover the material we have discussed since the first exam (Problem Sets 4,5,6), through class on Friday, October 27 (“Fermat’s Little Theorem”).

1) Functions/Mappings, injectivity and surjectivity

2) Integer division – know the statement of the Division Algorithm (Theorem 4.8 in the text) and its proof,

3) Greatest common divisors and Euclid’s algorithm for gcd(a, b). Know the proof of Theorem 4.20 – Every nonzero subgroup of the integers under addition has the form dZ for some d > 0, and how d satisfies the properties of a gcd (see Definitions 4.21 and 4.23).

3) Prime numbers, Euclid’s Lemma (Theorems 5.13 and 5.12), prime factorizations and the “Fundamental Theorem of Arithmetic (Theorems 5.17 and 5.18).

4) Congruence mod n and the integers mod n: the set of congruence classes Z/nZ and the addition and multiplication mod n operations.

Some Review Problems

From the course notes:

1) Chapter 4: 4.7, 4.9, 4.10

2) Chapter 5: 5.5, 5.8a, 5.9, 5.14, 5.15, 5.16

3) Chapter 6: 6.3, 6.4, 6.7, 6.9b,c

Review Session

I will be happy to run a review session before the exam. Tuesday evening (October 31) is probably the best time for this for me, especially if we do the exam on Thursday evening.

Sample Exam Questions

Note: This list is longer than the actual exam will be to show the range of different kinds of questions that might appear.

I. Let [a] ∈ Z/18Z and let m_{[a]} : Z/18Z → Z/18Z be the mapping defined by m_{[a]}([x]) = [a] · [x] = [a · x].

A) Is m_{[12]} an injective mapping? Why or why not? Is it a surjective mapping? Why or

why not? B) Is m_{[5]} an injective mapping? Why or why not? C) Prove that m_{[a]} is injective if and only if [a] has a multiplicative inverse in Z/18Z. D) Prove the m_{[a]} is injective if and only if m_{[a]} is surjective.

II. 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.

III. A) Prove that if S = ⟨a, b⟩ = {x ∈ Z : x = ma + nb, m, n ∈ Z}, then the smallest positive

integer d in S satisfies the properties:

1. d divides a and d divides b.

2. If c divides a and c divides b, then c divides d. B) Find the integer d = gcd(488, 376) and express d in the form d = m · 488 + n · 76 for

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

then gcd(a, n)|b.

IV. A) Show that for all n ≥ 1, 8|(9^{n} − 1). (Hint: Show first that 9^{k+1} − 1 = 9^{k} · 8 + (9^{k} − 1).) B) Restate the result of part A as a congruence. C) (Extra Credit-type question) Suppose we used base 9 rather than base 10 to represent

integers. How could you test numbers for divisibility by 8 using the base 9 digits? How could you test numbers for divisibility by 10 using the base 9 digits?

V. Show that [x] has a multiplicative inverse in Z/nZ (that is, a [y] such that [x][y] = [1] = [y][x]) if and only if gcd(x, n) = 1.

VI. A) Find all integer solutions of the congruence 12x ≡ 7 mod 331. B) Find all solutions [x] of the equation [17][x] + [4] = [2] in Z/29Z. C) Which [x] ∈ Z/18Z have multiplicative inverses?