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# Number Theory

### Riemann Sums, Taylor Polynomials, Taylor Residuals

Please see the attached file for the fully formatted problems. 1. ? Calculate the Taylor Polynomial and the Taylor residual for the function . ? Prove that as , for all . ? Find the Taylor series of f. ? What is the radius of convergence for the Taylor series? Justify your answer. 2. ? Let f:[0,1] be a bo

### Linear Equations and their Solutions

From the given polynomials, identify the polynomials of degree one. a. 311y - 5 - 43y b. (11y2)1/4 + 14 c. 10 + 19x - 2x2 d. 2 + 15x e. 52y4 + 7x + 2 f. (68)1y1 g. x3 + 3x - 9 Solve the following: i. -2x = 3x + 4 ii. 3x/4 = 6 iii. y/6 + 1 = 9 iv. 6 = -2x/4 v. Find f(1) for f(x) = 4x3 - 3x2 - x + 2

### Polynomials

A. Solve the following questions involving fundamental operations on polynomials a.Find p(x) + 4q(x) given p(x)=4x4 + 10x3 - 2x2 + 13 and q(x) = 2x4+ 5x2 - 3 b. Find P(-1/2) if P(x) = 2x4 + x3 + 12 c. Simplify: (-4 + x2 + 2x3) - (-6 - x + 3x3) - (-6y3 + y2) d. Add: (2x2 + 6y2 + 4z2 + 3xy + yz + zx) + (4x2 + 3y2

### Upper Level Number Theory Problems

Please see attached document. There are four problems. The first problem is 71! mod 73

### Euler Totient Function (Six Problems)

For this problem it helps to know that: 3x7x13 = 273 (a) Define the Euler Totient function, (SYMBOL) For (b) to (f) please see attached. (PLEASE SEE ATTACHMENT FOR COMPLETE PROBLEM AND PROPER SYMBOLS)

### Euclid's Algorithm for Greatest Common Divisor

1. (a) Use the Euclidcan Algorithm to find the greatest common divisor of 13 and 21 (b) Is 13 invertible in Z21? If so, find the reciprocal. (c) Suppose x and yare integers, what is the minimum positive value for 13x+21y? Determine all posible values of (x,y) for which the minimum is obtained. (PLEASE SEE ATTACHMENT FOR

### Factor Positive Integrers into Primes

Factor into primes the following positive integers: (a) 25 (b) 4200 (c) 10(to the exponent)10 (d) 19 (e) 1 *Please see attachment for proper citation and complete instructions

### A Discussion On Prime Factorization

Find two numbers that have a product of 81 and also have a sum of 30 (use prime factorization for the product) Please see attachment for the formatted question.

### Multiply and divide polynomials

Perform the indicated operation ... (see attachment)

### Polynomial Example Problems

Ace manufacturing has determined that the Cost of Labor for producing x transmissions is: 0.3x(squared) + 400x+550 dollars While the Cost of Materials is: 0.1x(squared) + 50x+800 dollars · Write a polynomial that represents the total cost of Materilas and Labor for producing x transmissions. · Evaluate the total cost pol

### Properties of complex numbers.

Given A=(-1/2+isqrt(3)/2)^n 1. Show that A is real for any natural n 2. Show that for n=3K where K is a natural number, A=2

### Complex Numbers Questions

1. The equation X^5 - 2X^4 - X^3 + 6X - 4 = 0 has a repeated root at X=1 and a root at X-2. By a process of division and solving a quadratic equation, find all the roots and hence write down all the factors of X^5 - 2X^4 - X^3 + 6X - 4 2. Given that cosX= (e^jx + e^-jx)/2

### Taylor Polynomials Using Substitution

F(x) = ln5 + ln(1-1/5x) Using substitution in one of the standard Taylor series, find the Taylor series about f for 0. Give all terms up to the term in x^3.

### Combinatorial and Computational Number Theory

See attached file for full problem description. (a) Prove that if g.c.d.(n,p) = 1,then p divides n^(p-1) -1. (b) Prove that if 3 is not a divisor of n, then 3 divides n^2 -1. (c) Pr

### Game Theory : Two-Player Card Game

In the two-player game of Two Stacks, a deck of cards (with the joker added, for a total of 53 cards) is randomly divided into two piles. The two players take turns removing cards from one pile or the other. On a player's turn, that player may remove any positive number of cards from a single pile. The object of the game is to r

### Synthetic Division, Long Division, Polynomial and Asymptotes

I am having trouble with the following problems. Can someone please help me and explain the process? P. 295 1. # 16 2. # 28 3. # 36 P.309 4. # 78 5. # 80 6. # 82 P.334 7. #28 8 #44 P.364 9. # 36 10. # 46(Use Grapher only) See the attached file.

### Proof : Polynomials

Problem: Let f(x) and g(x) be nonzero polynomials in R[x] and assume that the leading coefficient of one of them is a unit. Show that f(x)g(x) doesn't equal 0 and that deg[f(x)g(x)] = deg(f(x)) + deg(g(x))

### Product of two polynomials of smaller degree integer coefficient

The problem is attached. Express x^8 + 98x^4y^4 + y^8 as a product of two polynomials of smaller degree with integer coefficients.

### Number Theory

Find all the integers such that when the final digit is deleted the new integer divides the original one. Can you generalize this to deleting other digits?

### Cryptology Modular Arithmetic Functions

Decipher the following CFQGE KAZEMF ZMAGVMC NMO VYSV which was obtained by a formula of the type y=kx (mod 26)

### Explaining Problem Solving

A. Find 12/25 divide by 1/5 in two different ways. Explain your methods. b. Explain, using a diagram, how the following problems illustrate the two interpretations of division: partitive division ans measurement division. i. A road crew repaves 1 1/2 miles of road each day. How long will it take the crew to pave a 3/4 mile

### Working with Prime factorization

Rewrite in the simplest form. State the GCF(greatest common factor) of numerator and denominator in each case. 1. 34/85 2. 123123/567567 Find 5/9+7/12 using three different denominators. Give your answers as mixed numbers in lowest terms. State the LCM (least common multiple)of the denominators.

### Prime number proof

Prove that if p is a prime number, then p divides , for all n≥p. Here [r] denotes the greatest integer ≤ r , for any real number r. Does this result generalize to a result about instead of p ?

### Modular Arithmetic Functions

Show that z and iz have the same modulus. How are the graphs of these two numbers related?

### Expanding Polynomial Functions

Expand: (w+5)^4

### Simplify Polynomials Function

If m =/ 0, then (m4)/m4 = a)1 b)m2 c)m4 d)m12 e)m16

### Kernel equivalence and natural mapping

For a mapping %:A -> B, let == denote the kernel equivalence of %, and let *:A -> A== denote the natural mapping. Define \$:A== -> B by \$([a]) = %(a) for every equivalence class [a] in A==. 1. Show that \$ is well defined and one-to-one, and that \$ is onto if % is onto. Furthermore, show that % = \$*, so that % is the composite

### Approximation of Integrals and Taylor Polynomials

Given ( INTEGRAL ln square(x)dx, as x from n to n+1 ) = ( INTEGRAL ln square (n+x)dx, as x from 0 to 1 ) = ( INTEGRAL [[ln(n+x) - ln(x) + ln(n)]square] dx, as x from 0 to 1 ), (a) Verify that ( LIMIT (n/ln(n)) [INTEGRAL (ln square (x)dx) - (ln square (n))] as n approach to the infinity ) = 1 (b) Compute LIMIT ((n square)/

### Polynomial Function Characteristic Examples

Here is what the problem asks for: Give an example of a polynomial function f of degree 5 such that the only real roots of f(x) are -2,1,6 and f(2)=32. Show that your example works and leave f(x) in factored form.

### Approximation with Taylor Polynomials: Example Problems

Note:% just a symbol Let I be an open interval containing the point x.(x not), and suppose that the function f:I->R has a continuous third derivative with f'''(x)>0 for all x in I. Prove that if x.+h is in I, there is a unique number % = %(h) in the interval (0,1) such that f(x.+h) = f(x.) + f'(x.)h + f"(x.+%h)(h^2)/2