a) Show that 6, 28, 496, 8128, and 33550336 are perfect numbers (recall, according to the note: n is said to be perfect if σ(n) = 2n).

b) Recall that prime numbers of the form M_n := 2ⁿ − 1 are called the Mersenne primes. For those nsuch that M_n := 2ⁿ − 1 is prime,

prove that the number P_n := 1/2 (M_n + 1)M_n= 2^(n-1)(2ⁿ − 1) is a perfect number (Note: for P₁ = 6, P₂ = 28, P₃ = 496, P₄ = 8128, P₅ = 33550336 which recover the perfect numbers in (a)).

c) Let P = q · 2^(n-1) where q is an odd prime. Prove that if P is a perfect number, then q = 2ⁿ − 1, i.e. all perfect number of the form P = q · 2^(n-1) is of the form 2^(n-1) (2ⁿ − 1).

Find the order of each of the following elements.

( 3 , 4 ) in Z 4 × Z 6 ;

( 6 , 15 , 4 ) in Z 30 × Z 45 × Z 24;

( 5 , 10 , 15 ) in Z 25 × Z 25 × Z 25;

( 8 , 8 , 8 ) in Z 10 × Z 24 × Z 80.

Let ?=2^(2^?)+1 be a prime that n>1

1. Show that ? ≡ 2(mod5)

2. Prove that 5 is a primitive root modulo ?

For the following functions, determine if they are injective, surjective, bijective, or none and explain the reason. Recall that R: is the set of real numbers, Z: the set of Integers, and N: the set of natural numbers.

a. f: R → Z, where f(x) = ceiling(x)

b. f: Z → Z defined by f(x) = √5? + 1

c. f: Z → N defined by f(x) = |x + 1|

Situación:

Explicar el método para que una integral de línea sea independiente de la trayectoria. Ofrezca un ejemplo.

Translation

Situation:
Explain the method so that a line integral is independent of the trajectory. Offer an example.

I have to translate it into Spanish, if you know Spanish better, if not, well, I t

Situación:

Describa el método de establecer y evaluar para una integral triple. Ofrezca un ejemplo utilizando una de las siguientes opciones: (a) integral triple iterada, (b) uso de coordenadas esféricas, (c) uso de coordenadas cilíndricas.

Translation:

Situation:
Describe the method of establishing and evaluating for a triple integral. Give an example using one of the following options: (a) triple iterated integral, (b) use of spherical coordinates, (c) use of cylindrical coordinates.

If you can write it clearly it would be of great help, I need to hand it on Spanish, I'm going to translate it. Thanks!

(i) T(n) denote the number of distinct ways that a postage of n cents, where n ≥ 4 and n is even, can be made by 4-cent and 6-cent stamps. Find a recurrence relation T(n). NOTE [4,6] is the same as [6,4] so T(10) = 1 so T(n) is NOT T(n-4)+T(n-6)
(ii) Now assume we have 10-cent stamps in addition to the previous 2 kinds. Find a recurrence relation, S(n), for the number of distinct ways that a postage of n cents, where n ≥ 4 and n is even, can be made by 4-cent, 6-cent and 10-cent stamps.

An urn contains 8 white balls and 5 green balls. A ball is drawn at random, its color is noted, and

replaced together with 2 more of the same color. Then the selection is repeated one more time. What is

the probability that the ball in the second draw is green? ANSWER: (0.744)

Dont know how to get the answer. Would appreciate with the fewest amount of steps in order to get the answer.

The Taylors have purchased a $350,000 house. They made an initial down payment of $10,000 and secured a mortgage with interest charged at the rate of 6%/year on the unpaid balance. Interest computations are made at the end of each month. If the loan is to be amortized over 30 years, what monthly payment will the Taylors be required to make? (Round your answer to the nearest cent.) _____________

What is their equity (disregarding appreciation) after 5 years? After 10 years? After 20 years? (Round your answers to the nearest cent.)

5 years- $________

10 years- $________

15 years-$_________

In this question, we are going to call a function, f : R → R, type A, if ∀x ∈ R, ∃y ∈ R such that y ≥ x and |f(y)| ≥ 1. We also say that a function, g, is type B if ∃x ∈ R such that ∀y ∈ R, if y ≥ x, then |f(y)| ≥ 1.

Prove or find a counterexample for the following statements.

(a) If a function is type A, then it is type B.

(b) if a function is type B, then it is type A.

A two-lane tunnel opening in the shape of a semi-ellipse has a height of 16 feet and a width of 32 feet.

a) Determine an equation that represents the tunnel opening.

b) Explain with appropriate mathematical work and complete sentences whether a truck that has a height of 14 feet and a width of 8 feet can safely'enter the tunnel without crossing into the other lane of traffic and do not touch the sidewall. Assume the road is centered in the tunnel and the width of one lane is 12 feet.

Research a mathematician of your choice and write an essay about him/her (atleast 2 pages long, double spaced) You are expected to research the topic and write a clear, well-organized essay. The mathematician I've chosen is Blaise Pascal.

Use the method of undetermined coefficients (it works the same way for a fourth order linear equation) to find the general solution of:

y⁽⁴⁾ - 18y'' + 81y = 6t² - 4t + 36 + 90e^3t

[Hint: The complementary solution is y_c = C₁e^3t + C₂e^-3t + C₃te^3t + C₄te^-3t

If G={e, a, b, c, ...}. The set Inn (G)={phi e, phi a, phi b, phi c, ...} is called the set of inner automorphisms. Prove that Inn(G) is a group under the operation of function compositions.

BUSINESS TRAVEL EXPENSES An executive of Trident Communications recently traveled to London, Paris, and Rome. She  paid $280, $330, and $260 per night for lodging in London, Paris, and Rome, respectively, and his hotel bills totaled $4060. She spent $130, $140, and $110 per day for his meals in London, Paris, and Rome, respectively, and his expenses for meals totaled $1800. If she spent as many days in London as she did in Paris and Rome combined, how many days did she stay in each city? Solve using Gauz Jordan method.