7. Suppose Bob has the public key (n, e) = (21733, 691). You are Eve, and you have intercepted the ciphertext C = 21012. On a whim, you decide to check whether C and n are relatively prime, and to your delight, you discover that they are not! Show how you can use this to recover the plaintext M.

Note: The chance that M (or equivalently C) is not relatively prime to the modulus n is1/p + 1 /q− 1/pq , and when p, q are both larger than 100 digits long, this probability is less than 10^−99! So although this is a vulnerability that one should be aware of, in practice it doesn’t cause issues very often.

A study was conducted to determine whether a new drug, nomasbesos, was effective in preventing the swollen spleen that often occurs when a person has a mononucleosis (mono) infection. Individuals with mono were randomly selected to either receive nomasbesos or a placebo treatment. Among the 315 who received the nomasbesos, 35 developed swollen spleens. Among the 275 individuals who received the placebo, 92 developed swollen spleens. Go out to 3 decimal points for all your calculations and then round to 2 decimal points for your final answer. The answers are below for your studying purposes.

Complete the 2x2 for the study. Be SURE to indicate precisely what your treatment and outcome are and complete all applicable cells.

	Swollen spleen	No swollen spleen	Totals
Nomasbesos (treatment)
Placebo
Totals

1. What kind of study design (specifically) was used? How do you know?

2. What is the appropriate measure of association?

3. Calculate the value of the appropriate measure of association. Show all your work.

4. What does the calculated association mean in words?

Adirondack Savings Bank (ASB) has $1 million in new funds that must be allocated to home loans, personal loans, and automobile loans. The annual rates of return for the three types of loans are 5% for home loans, 13% for personal loans, and 8% for automobile loans. The bank’s planning committee has decided that at least 40% of the new funds must be allocated to home loans. In addition, the planning committee has specified that the amount allocated to personal loans cannot exceed 60% of the amount allocated to automobile loans.

(a)	Formulate a linear programming model that can be used to determine the amount of funds ASB should allocate to each type of loan to maximize the total annual return for the new funds. If the constant is "1" it must be entered in the box. If your answer is zero enter “0”.
	Let H = amount allocated to home loans P = amount allocated to personal loans A = amount allocated to automobile loans
	Max H + P   + A s.t. H    + P + A ≥ Minimum Home Loans H + P + A ≤ Personal Loan Requirement H +   P + A = Amount of New Funds
(b)	How much should be allocated to each type of loan?
	Loan type Allocation Home $ Personal $ Automobile $
	What is the total annual return?
	If required, round your answer to nearest whole dollar amount.
	$
	What is the annual percentage return?
	If required, round your answer to two decimal places.
	%
(d)	Suppose the total amount of new funds available is increased by $10,000. What effect would this have on the total annual return? Explain.
	If required, round your answer to nearest whole dollar amount.
	An increase of $10,000 to the total amount of funds available would increase the total annual return by $ .
(e)	Assume that ASB has the original $1 million in new funds available and that the planning committee has agreed to relax the requirement that at least 40% of the new funds must be allocated to home loans by 1%. How much would the annual return change?
	If required, round your answer to nearest whole dollar amount.
	$
	How much would the annual percentage return change?
	If required, round your answer to two decimal places.
	%

Suppose a water tower in an earthquake acts as a mass-spring system. Assume that the container on top is full and the water does not move around. The container then acts as a mass and the support as a spring, where the induced vibrations are horizontal. Suppose that the container with water has a mass of m = 10, 000 kg. It takes a force of 1000 Newtons to displace the container 1 m. For simplicity assume no friction. When the earthquake hits, the water tower is at rest (not moving). Suppose the earthquake induces an external force F(t) = A cos(ωt)

Find the solution to the Initial Value Problem

An element a in a ring R is called nilpotent if there exists an n such that aⁿ = 0.

(a) Find a non-zero nilpotent element in M2(Z).

(b) Let R be a ring and assume a, b ∈ R have a^t = 0 and b^m = 0 for some positive integers t and m. Find an n so that (a + b)ⁿ = 0. (You just need to find any n that will work, not the smallest!)

(c) Show that the set of nilpotent elements in a commutative ring R forms a subring of R.

(d) Does this subring from the previous question contain a unity?

(e) Are the Gaussian integers from an earlier question an integral domain? Explain your answer.

Each time draw 6 different numbers from 1 ~ 45 randomly

Let M denote the times you have to draw such that all the number from 1~45 has been drawn
What's the expected number of M.

please explain thoroughly. Thanks.

Find a basis and the dimension of the subspace:

V = {(x₁, x₂, x₃, x₄)| 2x₁ = x₂ + x₃, x₂ − 2x₄ = 0}

7.

(Please, if you are not willing to answer the question completely, please leave the question to someone who is!)

a. At the time of his daughter's birth, a man deposited $ 1,000 in an account that pays 6%; this amount is set every birthday. When he turned 12, he increased his appropriations to $ 1,500. Calculate the amount that will be available to her at age 18.

b. José earned $ 4,000,000 from the Puerto Rican lotus and will receive a check for $ 200,000 now and a similar one every year for 19 years. To guarantee these payments, the Electronic Lottery bought an anticipated annuity at the 10% interest rate compounded monthly. How much did the electronic Lottery cost the annuity?

Use MATLAB to figure out the following problem, if you do not know how to use MATLAB then please do not answer. Coding is required for the exercise.

For f(x) = arctan(x), find its zeros by implimenting Newtons method and the Secant method in Matlab. (Hint: Use Newtons method to calculate x1 for Secant method)

Comment all code please since I would like to learn how to do this correctly in MATLAB. Thank you.

Let ?1 ⃗ , ?2 ⃗ , ?3 ⃗ be three vectors from ℝ3 such no two vectors are parallel, and ?3 ⃗ is not in the plane spanned by ?1 ⃗ and ?2 ⃗ . Prove that {?1 ⃗ , ?2 ⃗ , ?3 ⃗ } forms a basis for ℝ3

Formulate the following problem using the following steps.

a. Define the decision variables.

b. Specify the objective function.

c. Specify the constraints and simplify them so that the left hand side of each constraint only contains terms involving the decision variables.

ChemLabs uses raw materials I and II to produce two domestic cleaning solutions A and B. The daily availabilities of raw materials I and II are 150 and 145 units respectively. One unit of solution A consumes .5 unit of raw material I and .6 unit of raw material II. One unit of solution uses .5 unit of raw material I and .4 unit of raw material II. The profits per unit of solutions A and B are $8 and $10 respectively. The daily demand for solution A lies between 30 and 150 units, and that for solution B between 40 and 200 units. Formulate and solve graphically the optimal daily production amounts of A and B

Nepal and Tibet can both produce butter [B] or tea [T].

        If Nepal allocates all resources to butter, it can produce a maximum of 500 units a year. If all its

        resources are allocated to tea, it can produce a maximum of 2500 units.

If Tibet allocates all resources to butter, it can produce a maximum of 1500 units a year. If all its resources are allocated to tea, it can produce a maximum of 3000 units of tea.

[a] Political considerations initially mean that trade is not possible between Nepal and Tibet.

        If Nepal produces 1250 units of tea for itself, how many units of butter does it produce for itself?

               

        If Tibet also produced 1250 units of tea for itself, how many units of butter does it produce for itself?

        What is the combined amount of tea produced by the countries when trade is not possible? The combined amount of butter?

[b] If political considerations change and trade becomes possible, which good will Nepal trade to

        Tibet? Prove your answer.

[c]   If Nepal and Tibet pursue complete specialization in the production of their comparative advantage products when initiating trade, what is the combined amount of tea produced by the countries?

        Under complete specialization in comparative advantage, what is the combined amount of butter?

        Comparing combined production prior to the possibility of trade [see [a]] with combined production available under specialization, what are the potential gains from trade as measured by additional butter and/or tea?  

[d] Assuming complete specialization by both countries, identify a specific trade [i.e. an amount of tea traded for an amount of butter] that will leave both countries better off when compared to their positions in [a].

Solve the following initial value problem: tdy/dt+5y=5t with y(1)=8.

Put the problem in standard form.

Then find the integrating factor, ρ(t)=

and finally find y(t)=

A) Solve the initial value problem:

8x−4y√(x^2+1) * dy/dx=0

y(0)=−8

y(x)=

B)  Find the function y=y(x) (for x>0 ) which satisfies the separable differential equation

dy/dx=(10+16x)/xy^2 ; x>0

with the initial condition y(1)=2
y=

C) Find the solution to the differential equation

dy/dt=0.2(y−150)

if y=30 when t=0

y=

Q: Find the power series solution centered at the ordinary point x = 0 for each equation.
((x^2) + 1) y''-6y=0