Let the cyclic group {[0], [1], [2], ..., [n − 1]} be denoted by Z/nZ. Consider the following statement: for every positive integer n and every x in Z/nZ, there exists y ∈ Z/nZ such that xy = [1]. (a) Write the negation of this statement. (b) Is the original statement true or false? Justify your answer.

Find the coordinates of the point (x, y, z) on the plane z = 4 x + 1 y + 4 which is closest to the origin.

Linear Programming

How do I use duality to find the optimal value of the objective function for this?

minimize 8y₁+6y₂+2y₃

constraints----

y₁+2y₂ ≥ 3

2y₁+y₃ ≥ 2

y₁ ≥ 0

y₂ ≥ 0

y₃ ≥ 0

NUMBER THEORY

1.Use the Euclidian algorithm to calculate the GCD of 1160718174 and 316258250.

2.Use Fermat’s Little Theorem to solve for x^86 ≡ 6 (mod 29).

Production Scheduling: New jet, Inc. manufactures inkjet printers and laser printers. The company has the capacity to make 70 printers per day, and it has 120 hours of labor per day available. It takes 1 hour to make an inkjet printer and 3 hours to make a laser printer. The profits are $40 per inkjet printer and $60 per laser printer. Find the number of each type of printer that should be made to give maximum profit and find the maximum profit using

1. SIMPLEX METHOD

Rolle's Theorem, "Let f be a continuous function on [a,b] that is differentiable on (a,b) and such that f(a)=f(b). Then there exists at least one point c on (a,b) such that f'(c)=0."

Rolle's Theorem requires three conditions be satisified.

(a) What are these three conditions?

(b) Find three functions that satisfy exactly two of these three conditions, but for which the conclusion of Rolle's theorem does not follow, i.e., there is no point c in (a,b) such that f'(c)=0. Each function should satisfy a different pair of conditions than the other two functions. For each function you should give a definition, a graph, and a short justification of its failing to meet the conclusion of Rolle's Theorem.

a) Find the recurrence relation for the number of ways to arrange flags on an n foot flagpole with 1 foot high red flags, 2 feet high white flags and 1 foot high blue flags.

b) solve the recurrence relation of part a

The Hastings Sugar Corporation has the following pattern of net income each year, and associated capital expenditure projects. The firm can earn a higher return on the projects than the stockholders could earn if the funds were paid out in the form of dividends.

The Hastings Corporation has 3 million shares outstanding. (The following questions are separate from each other).

a. If the marginal principle of retained earnings is applied, how much in total cash dividends will be paid over the five years? (Enter your answer in millions.)

b. If the firm simply uses a payout ratio of 20 percent of net income, how much in total cash dividends will be paid? (Enter your answer in millions and round your answer to 1 decimal place.)

c. If the firm pays a 20 percent stock dividend in years 2 through 5, and also pays a cash dividend of $3.40 per share for each of the five years, how much in total dividends will be paid?

d. Assume the payout ratio in each year is to be 40 percent of the net income and the firm will pay a 30 percent stock dividend in years 2 through 5, how much will dividends per share for each year be? (Assume the cash dividend is paid after the stock dividend.) (Round your answers to 2 decimal places.)

Consider the linear transformation T which transforms vectors x C) in the y-axis. a) Express the vector X = T(x), the result of the linear transformation T on x in terms of the components x and y of X. by reflection [10 Marks] b) Find a matrix T such that T(x) = TX, using matrix multiplication. Calculate the matrix product T2 represent? Explain geometrically (or logically) why it should be this. c) T T . What linear transformation does this a b d) Let A be a 2 x 2 matrix. Calculate the three matrix products TA, AT and TAT. For each, give a simple short description, in words concerning the rows and columns of A (say), of the result of the calculation to produce a new matrix from A.

Solve this problem with the revised simplex method:

Maximize            Z = 5X₁ + 3X₂ + 2X₃

Subject to            4X₁ + 5X₂ + 2X₃ + X₄ ≤ 20

                            3X₁ + 4X₂ - X₃ + X₄ ≤ 30

                           X₁, X₂, X₃, X₄ ≥ 0

Taxpayer T owns an office building worth $950,000, encumbered by a mortgage of $710,000. His original cost was $830,000, and he has taken depreciation deductions of $185,000 on the building.

T wants to exchange his building for another office building worth $800,000. He will assume the existing mortgage of $580,000 on the new building.

(This is a practice problem for an exam, please show work so I can understand the problem and how to get the correct answer.)

1. As stated, would this be a fair arms-length exchange? If not, who should be required to pay cash boot, and how much? Explain.
2. Assuming the exchange is made under the terms of your answer to #1, compute the following for T, showing all calculations:

a. Realized gain

b. Recognized gain

c. Basis in the new building

Subject: Combinatorics and Graph Theory

(Note: in any way could you possibly explain clearly step by step for this problem in what is being done. *Including what gadgets are used etc.)

*Problem: Could you reduce a 3-SAT to a Subset sum.

Let E/F be an algebraic extension and let K and L be intermediate fields (i.e. F ⊆ K ⊆ E and F ⊆ L ⊆ E). Assume that [K : F] and [L : F] are finite and that at least K/F or L/F is Galois. Prove that [KL : F] = [K : F][L : F] / [K ∩ L : F] .

Suppose V is finite-dimensional and S, T are operators on V . Prove that ST is bijective if and only if S and T are both bijective.

Note: Don’t forget that bijective maps are precisely those that have an inverse!