10.1 DO: LN 5.1.3, 5.1.17 (evaluate Netwon's binomial coefficients in terms of "old" binomial coefficients)

10.2 HW: LN 5.2.4 (a) - (e) (closed form expressions for generating functions)     (5+5+5+2+3 points)

10.3 DO: LN 5.2.5 (generating function of na_n)

10.4 DO: LN 5.2.6 (Fibonacci recurrence + 1 on the r.h.s.)

10.5 HW: Let a₀ = 1, a₁ = 1, a_n = a_n-1 + 6a_n-2.   (a) Find a closed-form expression for the generating function f(x) = ∑_n=0^∞ a_nxⁿ.   (b) Write this closed-form expression in terms of partial fractions. (c) Use the partial fractions to find an explicit formula for a_n.    (6+6+4 points)

10.6 DO: LN 5.2.8 (differential equation for generating function)

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In the problems below, F denotes a field and F[x] the set of polynomials in the variable x over F. Examples of fields include the rational numbers, real numbers, complex numbers, and the residue classes modulo primes. There are many other fields.

9.0 DO: A subset I of F[x] is called an ideal if (1) I is not empty; (2) I is closed under addition; (3) (&forall f ∈ I)(&forall g ∈ F[x])( fg ∈ I ). The principal ideal generated by the polynomial f consists of all multiples of f. (a) Observe that every principal ideal is an ideal. (b) Prove: every ideal in F[x] is principal.

9.1 DO: define gcd of polynomials over F. Prove gcd exists, is unique up to a nonzero contant factor, and can be written as a linear combination (what does this mean?).

9.2 DO: (unique prime factorization) Prove: Every nonzero polynomial over F can be factored into irreducible factors over F; this factorization is unique up to nonzero constant factros and the order of the factors.

9.3 DO: using the definition of the formal derivative of a polynomial, prove: (fg)' = f 'g + fg'.

9.4 DO: Recall that we say that a polynomial f has "multiple factors" if there is a nonconstant polynomial g such that g² | f. Prove: f has multiple factors over f if and only if f has multiple roots in the algebraic closure of F. [Added on Nov 10: Assume for this problem that the field has characteristic zero, i.e. the sum 1+1+...+1 (n times) is not zero for any positive integer n.] [For some, but not all, fields of characteristic p, the statement is false.]

9.5 HW: Prove: f has multiple factors over F if and only if gcd(f, f ') ≠ 1.    (12 points) [Added on Nov 10: This statements is also false for some, but not all, fields of characteristic p because 9.4 fails. Assume for this problem that the field has characteristic zero (see 9.4).]

9.6 DO: Prove: if f is a polynomial over the reals and α is a complex number and β is the complex conjugate of α then f(β) is the complex conjugate of f(α). Use this to prove that if α is a root of f then so is β.

9.7 DO: Prove: if a polynomial f is irreducible over the reals then deg(f) ≤ 2.

9.8 DEFINITION: Let F be a subfield of the field G. (Think of F being the rationals and G the complex numbers.) We say that α ∈ G is algebraic over F if there is a nonzero polynomial f over F of which α is a root. The monic polynomial of smallest degree with this property is the minimal polynomial of α over F. For instance, the minimal polynomial of 2^1/3 over the rationals is x³ - 2; over the reals, it is x - 2^1/3. The minimal polynomial of α over F is denoted m_α(x). (This depends on F.)

9.9 DO: Prove: m_α(x) is irreducible over F.

9.10 DO: Prove: (∀ f ∈ F[x])(f(α)=0 if and only if m_α | f ).

9.11 DO: Find the minial polynomial of the golden ratio (1 + √5)/2.

9.12 DO: Let f(x) = a₀ + a₁x + ... + a_nxⁿ and g(x) = a_n + a_n-1x + ... + a₀xⁿ. Assume a₀a_n ≠ 0.    f and g are called reciprocal polynomials. Prove: if α is a root of f then 1/α is a root of g with the same multiplicity.

9.13 REWARD PROBLEM: The roots of nonzero polynomials with integer coefficients are called algebraic numbers. Prove that the algebraic numbers form a field, i.e., sums, products, quotients of algebraic numbers are algebraic numbers. E.g., prove that 2^1/2 + 2^1/3 is an algebraic number.

9.14 REWARD PROBLEM: The roots of monic polynomials with integer coefficients are called an algebraic integers. Prove that the algebraic integers form a ring, i.e., sums and products of algebraic integers are algebraic integers.

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8.1 HW: Prove that there exists a constant C such that almost all graphs G=(V,E) on a given set V of n vertices satisfy the following statement:
 (*)    all vertices of G have degree between n/2 - C(n ln n)^1/2 and n/2 + C(n ln n)^1/2. (15 points)

8.2 CHALLENGE PROBLEM. Prove: there exists a constant C > 0 such that the statement (*) in Problem 8.1 is false for almost all graphs.

8.3 DO: Prove that there exists a constant c such that almost all graphs G=(V,E) on a given set V of n vertices satisfy the following statement:
 (**)    at least 99% of the vertices of G have degree between n/2 - c √n and n/2 + c √n.

8.4 DO: Prove that if 0 < c < 1/2 then for all sufficiently large n the probability that a random graph with n vertices is 3-regular (every vertex has degree 3) is less than exp( - c n²).   (Notation: exp(x) = e^x.)

8.5 HW: Let G be a random graph on the vertex set V={1,...,n}. Let p_n denote the probability that deg(1) = deg(2) (the first two vertices have the same degree). (a) Determine p_n. Give a simple closed-form expression. (b) Asymptotically evaluate p_n. Your answer should be of the form   p_n ∼ c n^d. Determine the constants c and d.    (8+4 points)

8.6 CHALLENGE PROBLEM. Prove: almost all graphs have Ω(√n) distinct degrees.

8.7 DO: LN 2.4.2 (asymptotic notation if limit of quotients exists)

8.8 HW: LN 2.4.3 (no limt but Θ)    (6 points)

8.9 DO: LN 2.4.4 (for positive sequences, a_n=Θ(b_n) implies   ln a_n= ln b_n + O(1))

8.10 DO: 2.4.5, 2.4.6 (addition vs O and Ω)

8.11 HW: LN 2.4.7 (a)(b) (does Θ imply log-asymp?)    (5+5 points)

8.12 DO: LN 2.4.9 (a)(b) (O(exp()) vs exp(O())

8.13 DO: LN 2.4.10, 2.4.11 (addition vs O and Ω again)

8.14 DO: LN 2.4.12 (harmonic sum = ln n + O(1))

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7.1 HW: Diseases A and B have similar symptoms. Let W be the population of all patients showing these symptoms. The two diseases can only be differentiated by costly tests. We know (from sampling the population and performing these costly tests) that 70 % of W have disease A, 25 % have disease B, and 5 % have some other disease. We consider the effectiveness of treatment T. We know that 60 % of the patients with disease A respond to T, while only 12 % of the patients with disease B respond to treatment T. From the rest of the population W, 40 % respond to treatment T.

(a) A new patient arrives at the doctor's office. The doctor determines that the patient belongs to W. What is the probability that the patient will respond to treatment T? (5 points)

(b) ("Probability of causes") The patient's insurance will not pay for the expensive tests to differentiate between the possible causes of the symptoms. The doctor bets on treatment T. A week later it is found that the patient did respond to the treatment. What is the probability that the patient had disease A? (Hint: review LN 7.1.9 ("Theorem of Complete Probability").) (10 points)

7.2 HW: LN 7.2.13 (club with 2000 members; assume club members must be of drinking age). Prove your answer. Make sure you give a clear definition of the random variables you use. The clarity of the definition accounts for 3/4 of the credit. (8 points)

7.3 DO: LN 7.4.2 (independence of events vs indicator variables). Use definition LN 7.4.1 of independence of random variables.

7.4 DO: LN 7.4.3 (equivalence of two definitions of indepedence)

7.5 HW (uncorrelatedness does not imply independence): Recall that the random variables X,Y are uncorrelated if E(XY) = E(X)E(Y). Construct random variables X,Y that are uncorrelated but not independent. Make your sample space as small as possible. You do not need to prove the minimality of your sample space.     (10 points)

7.6 HW (pairwise independence does not imply independence) THIS PROBLEM IS CANCELED because it was accidentally discussed during the Nov 3 tutorial. The problem was due Mon, Nov 9. REVIEW the solution; do not hand in.
Construct three random variables that are pairwise independent but not (fully) independent. Make your sample space as small as possible. You do not need to prove the minimality of your sample space. ("Fully independent" means the same as "independent;" the word "fully" is only inserted for emphasis. Pairwise independence means every two of the variables are independent.) Make your solution short; elegance counts.    (Originally announced value: 10 points.)

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6.1 HW (triangles in random graph): Let X denote the number of triangles in a random graph G = (V,E) on a given set V of n vertices. The adjacency relation is generated using (n choose 2) unbiased coin flips.   (a) What is the size of the sample space for this experiment?   (b) Determine E(X).   (c) Determine Var(X). Your answer should be a closed-form expression (no dot-dot-dots or summation or product symbols).   (d) Asymptotically evaluate Var(X). Your answer should be a formula of the form    c n^d. Determine the constants c and d.   (1+4+10+6 points)   posted Mon Nov 2 at 2pm.

6.2 DO: Prove: ∑_k=0ⁿ (n choose k)² = (2n choose n). (a) Give a combinatorial proof. (b) Give an algebra proof (use the Binomial Theorem).

6.3 DO: (a) Prove that the binomial coefficients (n choose k), k = 0,...,n, increase till the middle and then decrease.   (b) Let us consider a sequence of n independent Bernoulli trials with parameter p (biased coin flips with probability p of "heads"). Determine the probability p(n,k) that exactly k of the coins come up "heads." (c) Prove that the sequence p(n,0), p(n,1), ... , p(n,n) increases up to p(n,k^*) and decreases afterwards, where |k^* - np| < 1.

6.4 DO: LN 7.1.21 (asymptotically evaluate middle binomial coefficient)

6.5 DO: LN 7.1.5 (Union bound)

6.6 DO: LN 7.1.7 (conditional probabilities for intersection of three events)

6.7 DO: LN 7.1.9 (Theorem of Complete Probability) [this was previously erroneously listed as LN 7.1.19 - corrected Nov 3 at 12:30pm]

6.8 DO: LN 7.1.12 (independence of complement of event)

6.9 DO: Pick a random integer x ∈ {1,2,...,n}. Let A denote the event that x is even; and B the event that 3 | x. (a) For n = 1000, decide whether the events A and B are independent or positively or negatively correlated.   (b) REWARD PROBLEM. Decide for what values of n are A and B (i) independent; (ii) positively correlated; (iii) negatively correlated.

6.10 REVIEW def of independence of k events: LN 7.1.16

6.11 DO: LN 7.1.17 (three pairwise but not fully independent events)

6.12 DO: LN 7.1.18 (independence of functions of blocks of independent events)

6.13 DO: LN 7.1.19 (three-colored balls in an urn)

6.14 DO: LN 7.1.25 (if we have n nontrivial independent events, the sample space must have size ≥ 2ⁿ).

6.15 DO: LN 7.1.24 (k events that are (k-1)-wise but not fully independent)

6.16 CHALLENGE PROBLEM. LN 7.1.26 (small sample spaces for pairwise independent events)

6.17 CHALLENGE PROBLEM. LN 7.1.27 (lower bound for sample spaces for pairwise independent events) (Hint. Prove the same lower bound for pairwise independent non-constant random variables with zero expected value.)

6.18 DO: (a) Prove: if X is a random variable then min X ≤ E(X) ≤ max X. (b) If equality holds in either of these inequalities then what can we say about X ?

6.19 DO: Prove the linearity of expectation (LN 7.2.6)

6.20 DO: LN 7.2.9 (every random variable X is a linear combination of indicator variables which are functions of X)

6.21 DO: 7.2.12 (lottery)

6.22 DO: 7.2.14 (expected number of edges of a graph)

6.23 DO: 7.2.16 (randomly stuffed envelopes) (solved in class; review)

6.24 REVIEW LN Sec 7.3 (Chebyshev's inequality)

6.25 DO: LN 7.3.10 (Cauchy-Schwarz)

6.26 DO: Show that Chebyshev's inequality (combined with the union bound) is not strong enough to solve LN 7.3.9.

6.27 CHALLENGE PROBLEM: LN 7.3.11 (limit on negatively correlated events)

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5.1 DO: LN 4.2.20 (Euler-Fermat congruence). Hint: adapt the proof of Fermat's little Theorem given in class.

5.2 DO: Given a and m we want to find k > 0 such that a^k ≡ 1 mod m. Prove: such k exists if and only if a and m are relatively prime.

Recall from class that the order of a modulo m is the smallest k > 0 such that a^k ≡ 1 mod m. Notation: k= ord_m(a). By Exercise 5.2, ord_m(a) is defined if and only if gcd(a, m) = 1.

5.3 DO: Prove: If a^t ≡ 1 mod m then ord_m(a) | t. Note in particular that ord_m(a) | φ(m); in particular, ord_p(a) | p - 1 (p a prime).

5.4 HW: Determine the order of 8 modulo 209 (209 =11.19). Prove your answer. Do as little calculation as possible; elegance counts.       (10 points)

Recall from class that g is a primitive root mod p (p prime) if gcd(g,p) = 1 and ord_p(g) = p - 1.

5.5 DO: Let p be a prime. Assume gcd(g,p) = 1. Prove that the following statements are equivalent:

• g is a primitive root mod p
• the numbers 1, g, g², ... , g^p-2 are pairwise not congruent mod p
• (∀a)(if gcd(a, p) = 1 then (∃x)(g^x ≡ a mod p))    (the "discrete logarithm" problem has a solution)

5.6 DO: Find a primitive root modulo each prime p < 20.

5.7 DO: Determine the number of primitive roots mod p. Your answer should be a very simple formula involving an arithmetic function we have studied.

5.8 DO: Prove: (a) If 2^k - 1 is prime then k is prime (Mersenne primes) (e.g., 31 = 2⁵ - 1) (b) If 2^m + 1 is prime then m is a power of 2 (Fermat primes) (e.g., 17 = 2²² + 1).

5.9 CHALLENGE PROBLEM. Prove: if p^t divides the binomial coefficient (n choose k) then p^t ≤ n. (p is a prime.)

5.10 DO: Prove: the graph G is a tree if and only if there is a unique path between every pair of vertices.

5.11 DO: Prove: every tree is bipartite.

5.12 DO: Prove: a graph G has a spanning tree (a spanning subgraph that is a tree) if and only if G is connected.

5.13 DO: LN 6.1.29 (good characterization of bipartite graphs) (For an elegant proof, use 5.10.)

5.14 HW: LN 6.1.37 (Grötzsch's graph.) Give a nice drawing of your graph.     (8 points)

5.15 CHALLENGE PROBLEM. LN 6.1.38 (triangle-free graphs with large chromatic number)

5.16 HW: LN 6.1.35 (chromatic number vs. independence number)     (8 points)

5.17 DO: LN 6.1.43 (Hamiltonicity of the grid)

5.18 HW: Find a graph G with a maximal independent set S such that α(G) = 100|S|. (An independent set is maximal if it cannot be extended to a larger independent set. You are asked to show that maximal independent sets can be quite small compared to maximum independent sets, i.e., independent sets of maximum size.)     (8 points)

5.19 DO: LN 6.1.33 (max-deg + 1 colors suffice)

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4.1 DO: Let p be a prime. Prove that Fermat's little Theorem is equivalent to the statement that
       (∀a)(a^p ≡ a mod p).

4.2 HW: Let p be a prime number. (a) Prove: (a + b)^p ≡ a^p + b^p mod p. (Hint: use HW 3.12.)   (b) Prove the 4.1 version of Fermat's little Theorem by induction on a.   (4+6 points)

4.3 DO: Let p be a prime. Prove: if p^k | n! then k < n/(p - 1).

4.4 HW: LN 5.1.1 (inductive proof of binomial sum)       (5 points)

4.5 HW: LN 6.1.10 (if G is a bipartite graph then m ≤ n²/4).       (6 points)

4.6 CHALLENGE PROBLEM: LN 6.1.11 (Mantel - Turán)

4.7 HW: LN 6.1.5(a)(b)(c) (self-complementary graphs)       (1+1+8 points)

4.8 HW: LN 6.1.16 (draw all trees with 7 vertices)       (9 points; lose 2 points per mistake)

4.9 REWARD PROBLEM. Verify Cayley's formula for n = 7 by computing the number of automorphisms for each tree in problem 4.7.

4.10 DO: LN 6.1.21 (count 4-cycles in complete bipartite grphs)

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3.1 DO: LN 2.2.1 - 2.2.5 (basic properties of asymptotic equality)

3.2 DO: LN 2.2.6 (a) (by Friday) 2.2.6 (b)(c)(d) (by Monday, Oct 19) (basic asymptotic equalities)

3.3 HW: Let {a_n} and {b_n} be sequences of positive numbers. Consider the following statement: "If a_n ∼ b_n then ln a_n ∼ ln b_n."
(a) Assume a_n > 1, b_n > 1. Prove that the statement is false.
(b) Assume a_n → ∞. Prove that the statement is true.       (6+8 points)

3.4 HW (due Monday, Oct 19): LN 2.2.7 (find a_n ∼ b_n such that a_nⁿ ≁ b_nⁿ)     (8 points)

3.5 DO: LN 2.2.9 (asymptotically evaluate the middle binomial coefficient)

3.6 DO (due Monday, Oct 19): LN 2.2.10 (simple proof of the asymptotics of ln(n!))

3.7 DO (due Monday, Oct 19): LN 2.2.12 (asymptotics of the n-th prime)

3.8 DO: LN 2.2.13 (chance of picking a prime)

3.9 DO (due Monday, Oct 19): LN 2.3.2 - 2.3.7 (basic properties of the little-oh notation)

3.10 DO: LN 5.1.1 (inductive proof of binomial identity)

3.11 DO: LN 5.1.4 (number of odd subsets = number of even subsets: two proofs)

3.12 DO (due Monday, Oct 19): LN 5.1.2 (p divides binomial coefficient)

3.13 DO: LN 5.1.5 (count even subsets)

3.14 REWARD PROBLEM: LN 5.1.6 (count sets of size divisible by 4)

3.15 HW (due Monday, Oct 19): LN 5.1.9 (lower bound for binomial coefficients)     (6 points)

3.16 HW (due Monday, Oct 19): LN 5.1.10 (upper bound for binomial coefficients)     (10 points)

3.17 CHALLENGE PROBLEM: LN 5.1.11 (upper bound on tail of binomial series)

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2.1 DO (due Monday, Oct 12): Solve Quiz 1. (On the class home page, click "grading, tests" on the banner and then click "first quiz.")

2.2 DO LN 4.1.16 (Euclid's algorithm is fast)

2.3 HW: LN 4.1.18 (solve system of simultaneous congruences) Show all your work! Explain how you found the solution. (8 points)

2.4 HW: LN 4.1.19 (decide whether or not system of simultaneous congruences is solvable; if YES, solve, if NO, prove) (8 points)

2.5 DO: LN 4.1.20 (system of simultaneous congruences)

2.6 DO: LN 4.1.21 (gcd vs congruence)

2.7 DO: LN 4.1.22(b) (Fibonacci numbers, induction)

2.8 DO: LN 4.1.22(d) (gcd of powers minus one)

2.9 HW: LN 4.1.23 (large power of 3 modulo 173) Show all your work! Lose points for unexplained steps. Do not use calculators for anything other than multiplication and division of numbers with no more than six digits. Indicate in what steps you used a calculator. (8 points)

2.10 HW: LN 4.1.24 (a)(b)(c) (square roots of 1 mod m) (4+4+12 points)

2.11a HW: LN 4.1.25 (square root of negative 1 mod 419) (The solution should be very short - essentially one line) (8 points)

2.11b HW: LN 4.1.26 (a³³≡a (mod 85)) (8 points)

2.12 DO: LN 4.1.27 (1-4)

2.13 DO: LN 4.1.28 (1-3)

2.14 DO: LN 4.1.29 (a³⁷≡a (mod 247=13.19))

2.15 DO: LN 4.1.32 (a) (n+1 numbers from {1,...,2n})

2.16 CHALLENGE PROBLEM: LN 4.1.32 (b)

2.17 CHALLENGE PROBLEM: LN 4.3.24

2.18 DO: (a) read LN 4.3.1 (def of Euler's φ function) (b) LN 4.3.3 (sum of φ(d) over divisors of n)

2.19 DO: (a) read LN 4.3.25 (def of multiplicative arithmetic functions) (b) Prove that Euler's φ function is multiplicative (c) LN 4.3.27 (1) (φ of prime powers) (d) LN 4.3.29 (formula for Euler's φ function)

2.20 CHALLENGE PROBLEM: LN 4.3.4 (determinant of gcd's)

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Homework marked for in RED for Wednesday was originally intended for Monday.

1.1. DO: review the concept of limits, especially at infinity. Define them using properly quantified formulas.

1.2 (due Fri, Oct 9) STUDY asymptotic equality (LN Chap 2.2)

1.3. DO: Prove: every equivalence relation arises from a partition. (Two elements are equivalent if and only if they belong to the same block of the partition.)

1.4. DO: (a) When are two integers congruent modulo 1? (b) When are two integers congruent modulo 0? (c) List the residue classes modulo 4 (by representatives). Construct their addition and multiplication tables. (These will be 4 by 4 tables with entries 0,1,2,3.) (d) Prove: every prime number other than 2 is congruent to 1 or -1 modulo 4. (e) Prove: every prime number other than 2 and 3 is congruent to 1 or -1 modulo 6.

1.5. DO: Prove: if a | x and a | y then a | x+y and a | x-y. (The vertical bar stands for divisibility: "a divides x," etc.)

1.6. HW:(Due Wed, Oct 7) LN 1.2.5 (a)(b)(c) (5+5+5 points) (Numbers and quantifiers) Give clear TRUE/FALSE answers to each problem and prove your answers. The proofs should be very short.

1.7 DO: LN 4.4.22 (sum of two squares never -1 mod 4)

1.8. HW: (a) Prove: if x ≡ -1 (mod 8) then x cannot be written as a sum of 3 squares. ("≡" stands for congruence.) (b) Write this statement as a properly quantified expression. (10+2 points)

1.9. DO: LN 4.1.15 (a)(b) (Number of divisors)

1.10. HW:(Due Wed, Oct 7) LN 4.2.12 (10 points) (Periodicity of the Fibonacci numbers modulo m).

1.11. DO: Study Euclid's Algorithm (LN Sec 4.1)

1.12. HW:(Due Wed, Oct 7) Let DIV(x) denote the set of divisors of x. For instance, DIV(6) = {± 1, ± 2, ± 3, ± 6}. (a) What is DIV(0) ? (b) Prove: DIV(x) ∩ DIV(y) = DIV(x-y) ∩ DIV(y). (2+10 points)

1.13. HW (due Wed, Oct 7): LN 4.1.15 (c) (10 points) (Upper bound on the number of divisors)

1.14 DO: LN 1.2.5 (d)(f)(h) (logic and numbers)

1.15 DO (due Tuesday, Oct 6, before the tutorial): LN 4.1.27 (logic and gcd)

1.16. HW (due Wed, Oct 7): Prove: if d is a common divisor of a and b and at the same time d is a linear combination of a and b then d is a greatest common divisor of a and b. (10 points)

1.17. HW (due Wed, Oct 7) LN 4.2.9 (members in a mod m residue class have the same gcd with m) (8 points)

1.18. DO (Due Tue, Oct 6, before tutorial): LN 4.14 (a) (b) (compute gcd's using Euclid's Algorithm). In addition, find a representation of the gcd's obtained as linear combinations.

1.19. HW (Due Wed, Oct 7) LN 4.1.17 (a)-(e) (multiplicative inverse) (2+5+5+5+5 points) (This was mistakenly posted as LN 4.1.16 (which does not have parts (a)-(e)). Corrected Wed, Oct 5, 2pm.

1.20 (due Fri, Oct 9). REVIEW mathematical induction.

1.21. HW (due Fri, Oct 9) LN 4.1.22 (a) (consecutive Fibonacci numbers are relatively prime) (8 points)

1.22. DO (due Fri, Oct 9): LN 4.1.21 (gcd vs congruence mod 5)

1.23. DO (due Fr: LN 4.1.27 (logic and gcd)

1.24. DO (due Fri, Oct 9): (a) Define the least common multiple in analogy with our definition of the greatest common divisor.

(b) Prove that l.c.m. exists. Hint: do not try to mimic the proof of the existence of a g.c.d.; use instead the fact that g.c.d. exist: represent the l.c.m. as the g.c.d. of a certain set of numbers (which numbers?).

(c) LN 4.2.10 ( gcd(a,b)lcm(a,b)= |ab|.)

1.25. DO (due Fri, Oct 9): LN 4.1.27 (tiny questions on gdc, lcm, congruence mod 24)

1.26. CHALLENGE PROBLEM. LN 4.3.4. (Determinant of gcd's.)

1.27. CHALLENGE PROBLEM. LN 4.2.13 (Integer-preserving polynomials)

1.28. CHALLENGE PROBLEM. LN 4.2.2, part 2. (Divisor game)

1.29. REWARD PROBLEM: LN 4.2.6 (gcd of powers of a minus 1)

1.30. CHALLENGE PROBLEM (no point value but earns instructor's attention; no deadline): LN 4.2.8 (gcd of Fibonacci numbers)

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