*** Let R be a unique factorization domain. Let p E R. If x is irreducible, then it is prime.***

Q: Prove that there is only one polynomial P3(x) among all polynomials of degree < 3 that satisfy the…

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Q: Number Theory

A: Since we only answer up to 3 sub-parts, we’ll answer the first 3. Please resubmit the question and…

Q: Prove that there exists exactly one prime of the form n^3 -1. (When we say that a number has "the…

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Q: 7. Give a non-combinatorial proof of the following identities: (a) n - k = n k - 1 (b) EOC)-)- n-k m…

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Q: Exercise 3. Give a combinatoric argument to prove the identity 2 2n (3.1) j=0 Hint: A subset of n…

A: Let the total persons selected be n out of 2n persons  Of which let male selected be j and the women…

Q: 5.2.8 suppose that a store offers gift certificates and denominations of $25 and $40. Determine the…

A: The given denominations are $25 and $40.Here, the given denominations are multiples of 5.The amounts…

Q: do I prove this: Let a, b ∈ Z. Give an example of an integer x so that x∣a and x∣b and If x∣a and…

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Q: 8. Prove using induction that E-oG) = 2 1 k=0 - 2n

A: We use the principle of mathematical induction to prove the given statement.

Q: It states that if an input is multiplied by the factor A, the output will be multiplied also by the…

A: LTI theorem follows that if a system is both linear and time-invariant now since the system is…

Q: Determine whether 3 and 5 are irreducible or not in Z[i]. 3 and 5 are reducible. O 3 and 5 are…

A: We have to check

Q: What is the characteristic polynomial for the recurrence relation an = 5an-1 –7.2 (1) (x + 5)(x – 1)…

A: Given recurrence relation an=5an-1-7

Q: 2. Let U = (3,7]. %3D a. Prove that 3.1 is not the greatest lower bound for U. b. Prove that 2.9 is…

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Q: true or false In the domain of Gaussian integers Z[i], the element 17 is Irreducible

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Q: 3.2. Prove the Pigeonhole Principle (Example 3.19) by induction on |Y| instead of on \X}. Example…

A: We will use basic theory of logic and fucntions.

Q: Prove that gcd(k; n) = gcd(n - k,n).

A: Greatest common divisor (gcd) :            If a , b are two integers such that atleast one different…

Q: 3. Let N eNand let PN be the set of all polynomials in the variable x of degree at most N with…

A: For the solution follow the next step.

Q: Let k > 2 be an integer and let n1, n2, ... , Ns E N be pairwise coprime positive integers. Show…

A: We prove given statement by mathematical induction in 3 steps. Step 1: We prove given statement for…

Q: (a) Find all the irreducible representations of S3.

A: To Determine :-  All the irreducible representations of S3.

Q: 1. Let q-p" be a prime power. Show that X+ 1 is irreducible in F [X] if and only if q=3 mod4. (Hint:…

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Q: Prove that there exists exactly one prime of the form n° -1. (When we say that a number p has "the…

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Q: Determine all integers n ≥ 3 such that Cn is planar.

A: ANSWER :

Q: Coefficient of ' in the expansion of (1+x) (1+ x)* is

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Q: all odd degree terms in the expansion of (x + v(x³-1)5+ (x – v(x³-1)5, (x > 1) is

A: Introduction :

Q: 1) Let x be a positive real number. Show by induction on n that (1+x)" > 1+ nx for all integers n >…

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Q: The number of irreducible monic polynomials of degree 2 over Z5 O 10 15 O 21 O None of the choices O…

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Q: The specialization of structural induction for ______ or  _______is called mathematical induction.

A: That's easy.

Q: 9. lim xln (1 + (Hint: Indeterminate Product)

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Q: 5)ohow all Stes in the solution of the polynomial ineguality beiow. (メーム じ o

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Q: 4. If G = (Z8,) then G has an element of order 5. *True *False

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Q: Obtain the expansion for (3 + 2x)* in ascending powers of x

A: According to binomial expansion,  a+bxn=an+n1an-1bx+n2an-2b2x2+n3an-3b3x3+....+nn-1abn-1xn-1+bnxn…

Q: (a) Show that the number of partitions of 10 into distinct parts (integers) is equal to the number…

A: In part a, we have to find how many possible partitions are there for 10, then find how many odd…

Q: Prove gcd(n, n+1) = 1.

A: To prove the below expression. gcdn, n+1=1

Q: 5.2.8 More generally, use induction to prove that (a - b) | (a" – b") for any positive integers…

A: Given: (a-b) | (an-bn) ----(1) Substitute n=1 in equation (1): (a-b) | (a1-b1)=1 The assertion is…

Q: S = {x2 − y2 ∈ N : x, y ∈ Z, x2 − y2 > 127}. Prove that S has a smallest element.

A: Given S={x2-y2∈N: x,y∈Z, x2-y2&gt;127} Theorem (Well ordering Principle) Every Nonempty subset of…

Q: Show that O let @ (x) be the step funetion Show that b x

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Q: 7. Construct the generating functions for the number of partitions of n such that: a. No part is a…

A: Determine the generating functions for each part. Write the generating function of part (b) and then…

Q: a1 = 1, an+1 = 3 - An induction method to prove monotor

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Q: 8. All polynomials in P, such that p(0) = 0.

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Q: 2. Let U = (3,7]. a. Prove that 3.1 is not the greatest lower bound for U. b. Prove that 2.9 is not…

A: Given problem is :

Q: What is Wronskian of e' and et? 1. -2 -2t O000

A: Given that  e^t and e^(-t)  To find the wronskian of e^t and e^-t

Q: 7. Use the binomlal theorem to expand (x+x-1] and simplify the result. Show your work.

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Q: The number of irreducible monic polynomials of degree 2 over Z5 O 21 15 O None of the choices O 10

A: We know that there are exactly (p2-p)/2 irreducible monic polynomial of degree 2 over Zp.

Q: The number of irreducible monic polynomials of degree 2 over Z7 is: * O 15 None of the choices 21 10

A: Since you have asked multiple question, we will solve the first question for you. If youwant any…

Q: 1. Find the first-order derives:

A: Given,      4x2+715x4+534x3-37x6+15x-30

Q: Show that x4 – 22 x2 +1 is irreducible over ℚ. [Hint: Show that it has no factor of degree 1 and…

A: Given the given expression is x4-22x2+1. Use theorem 23.11 and corollary 23.12.

Q: Let k, m, n be positive integers with k <m <n. Give a combinatorial proof of the identity k m m+ Σ k…

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Q: Find the term of free x in the 10 2 expansion of (x² + 3)¹0.

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Q: 5. Show that there are infinitely many irreducible polynomials in F,[X].

A: By the method of contradiction i have solved the problem

Q: Let а, b be positive coprime integers. a) Show that {0, a, 2a, . , (b – 1)a} forms a complete…

A: Congruent modulo : For any two integers p and q if p-q is divisible by a fixed positive integer k ,…