### Congruences

5. A polynomial is said to be monic is its leading coefficient is perpendicular... Please see attached.

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5. A polynomial is said to be monic is its leading coefficient is perpendicular... Please see attached.

Problem 1. Let n be a natural number and a1.... ,an > 0 be pairwise different positive real numbers. Show that if λ1...λn are such real numbers that the equality ... holds true for all x E R then .... Problem 2. Show that there are infinitely many real numbers x in the interval [0, pi/2] such that both sinx and co

Proofs 1. Let F be a field and p(x) and irreducible polynomial over F. Prove that (p(x)) is a prime ideal of F[x]. 2. If R has no divisors of zero, then neither does R[x].

I hope you will be able to help me with this problem, I'm really stuck. 10-(k+5) = 3(k+2) I would also appreciate any explanation.

Write 4^-2 x √16 x 3/12^-1 as powers of prime factors.

Recall that a perfect sqaure is a natural number n such that n = (k^2), for some natural number k. Theorem. If the natural number n is not a perfect square, then n^(1/2) is irrational. Proof. S(1): Suppose n^(1/2) = r/s for some natural numbers r and s. S(2): We may assume that r and s have no prime factors in common,

1. Prove the following lemma. Lemma Suppose that m and n are natural numbers > 1 and that p is a prime number. The following statements are equivalent: a. p is a prime factor of m or p is a prime factor of n. b. p is a prime factor of m*n Also Use Theorem: The Fundamental Theorem of Arithmetic. 2. Prove the follow

Is there a perfect square n^2 such that n^2 = -1 (mod p) for p=3 p=5 p=7 p=11 p=13 p=17 p=19? Can you characterize the primes for which n^2 = -1 (mod p) has a solution?

Let X and Y be independant random variables that are both equally likely to be either 1,2... (10)^N, where... a) Give a heuristic argument that Qk = 1/k^2Q1. (See attachment for full questions)

For any set B, let P(B) denote the power set of B (the collection of all subsets of B): P(B) = {E: E is a subset of B} Let A be a countably infinite set (an infinite set which is countable), and do the following: (a) Prove that there is a one-to-one correspondence between P(A) and the set S of all countably infinite seq

Write the numbers 25, 32, 56 to the base 5. write 47, 68, 127 to the base 2.

Please see attached for question

B6: a) State a formula for tau (n), the number of divisions of n, in terms of the collected prime factorization of n. b) define the term multiplicative function. c) Suppose that f and g are multiplicative functions. Prove that the function h defined by h(n) = SUM (d|n) f(d)*g(n/d) is also multiplicative. d) Fi

When solving a quadratic equation using the quadratic formula, it is possible for the b2 - 4ac term inside the square root (the discriminant) to be negative, thus forcing us to take the square root of a negative number. The solutions to the equation will then be complex numbers (i.e., involve the imaginary unit i). Question:

1. FIND ALL VERTICAL ASYMPTOTES OF THE FUNCTION. *******X+5 F (x) ------------------- ****4x squared+7 2. WHICH SHOWS THE TRUE STATEMENT FOR THE GRAPH OF THE RATIONAL FUNCTION g. *******X+2 g(x) ------------------- ****x squared+2x-3 3.USE SYNTHETIC DIVISION TO FIND UPPER AND LOWER BOUNDS OF THE REAL ZEROS OF f.

1. Identify the polynomial written as a product of linear factors. F(x) = x fourth +10x cubic +35x squared +50x +24 2. Solve z cubic-6z squared + 13z-10 given that 2 + i is a root. 3. Find a polynomial with interger coefficients that has the given zeros. 5, 4i, -4i, i, -i 4. Find all vertical asymptotes of th

1.USE DESCARTES` RULE OF SIGNS TO DETERMINE THE POSSIBLE NUMBER OF POSITIVE REAL ZEROS OF THE FUNCTION. f(x)=3x cube-4x squared-2x-4 ANSWER: A.3 or 1 B.1 C.2 or 0 D.5, 3, OR 1 2.USE SYNTHETIC DIVISION TO COMPLETE THE INDICATED FACTORIZATION:x cube-7x-6=(x-3)( ) 3.USE SYNTHETIC DIVISION TO COMPLETE THE INDICA

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

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

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

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)

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 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

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.

Perform the indicated operation ... (see attachment)

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

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

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.

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

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