Which one of the following is true? In the ring Z6, if x = x then just x = 0 or x = 1 In the ring Z6, 4 does not divide 2 In the ring Zp, if xy = 0 then x = 0 or y = 0 %3D In the ring Z8, if xy = xz and x + 0 then y = z

Q: Find the number of elements in the conjugacy class of (abc)(def) in S₆

A: We have to find the number of elements in the conjugacy class of (a b c)(d e f) in S₆

Q: Q6: Define the Boolean ring. Is (Z,+,.) Boolean?

A: .

Q: Give and write 5 example each of polynomials with degree 3, 4, 5, and 2.

A: If a polynomial is degree n then highest degree should be n with non zero coefficient

Q: 3. If p = 4x3 + x² + 2x + 5 and %3D q = 6x3 + 3x² + x + 2 Find the product in p and q, ||p|| and…

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Q: Could you please help me out with this question?

A: In the ring of integers modulo  n (Z/n), the multiplicative inverse of an integer a  exists if and…

Q: Preview Activity 8.4.1. Preview Activity 8.3.1 showed how we can approximate the number e with…

A: The given function is f(x)=ln(1+x). (a) To Find: Linearization of the function and then approximate…

Q: Determine if the Sóllowing justi fy by egprvelent: 1. Ore showng work. and ムx-2 2

A: Given:  12x2 - 4x / 2 and  6x - 2

Q: 3. Show that Z10 is isomorphic to Z2 × Z5, but that Zg is not isomorphic to Z₂ × Z4.

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Q: Suppose N = Fox F₁x F2, where the three factors are pairwise prime; hence, each factor and the…

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Q: find the generated of zy zs+Z7

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Q: The number of zero divisors of the ring Z, O Zg is O 1 O 5 None of these O 7

A:

Q: If x is divisible by 5 or y is divisible by 5, then xy is divisible by 5.

A: Let x is divisible by 5. Then x=5u for some natural number u.

Q: Show that the the greatest common divisor gcd(16" – 4",17" – 2") is divisible by 3 for all n E N.

A: Consider the provided question,

Q: Prove by induction that 32n 1 is divisible by 8 for Vn 20

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Q: If x2 + 9x + 14 х2 — 49 is divided by Зх + 6 x2 + x – 56 , the quotient is:

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Q: how the element 2+√3 has a multiplicative inverse in Q(√3). Show that Q(√3) is a subfield of C

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Q: List all zero-divisors in Z20. Can you see a relationship between thezero-divisors of Z20 and the…

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Q: The number of zero divisors in the ring Z80

A: Step:-1 Theorem:- If  gcda, n>1  and n- a then [a] is a zero divisor in ℤn. Also, we know ϕn =…

Q: Suppose A = {1, 2, 4, 5}, B = {2, 3, 5, 6} & C = {4, 5, 6, 7}. (a) What is (A ∪ B) − (C − B)?(b)…

A: We have given A = {1, 2, 4, 5}, B = {2, 3, 5, 6} & C = {4, 5, 6, 7}.

Q: a'b in (a + b)'07

A: a+b10 Now the general term is, Tr+1=Crn·an-r·br⇒Tr+1=Cr10·a10-r·br

Q: Use induction to prove that the product of any three consecutive positive integers is divisible by…

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Q: Cameron's work is shown below. Which of the following statements is correct? * 22+ 4x-3 x+5 A.) This…

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Q: We have to show that nonzero elements in Zp where p is prime have multiplicative inverses

A: We have to solve given problem:

Q: The characteristic of the ring 6x8 ×; is 14 (A) 2 (B) 48 (C) 84 (D) 168

A: We have to find the characteristic of the ringWe know thatCharacteristic of a Ring : Let R be a…

Q: What is the coefficent of 5 in (x + 1)7 - x²(x+3) 5

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Q: Utilize the method of mathematical induction. Verify: 1 1x3 3x5 5x7 y 1 (2y-1)(2y+1) 2 +1 for every…

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Q: Suppose that |a| = 24. What is the generator of (a²l)n (a!o)? a20 a a14

A: Suppose that |a| = 24. We have to find the generator of <a21>∩<a10> .

Q: Let x € Z. Prove that if x² - 8x + 11 is divisible by 4, the x6- 1 is also divisible by 4.

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Q: Let x Є Z. Prove that if x2 - 8x + 11 is divisible by 4, the x6-1 is also divisible by 4.

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Q: 3. Find the greatest common divisor of: (a) The polynomials r³ + x² + x and r³ +x* +x² + x in Z2.

A: The given polynomials are: x3+x2+x, x5+x4+x2+x The given group is ℤ2.

Q: What is the coefficient of x 9 in (x 3 + x) 5?

A: What is the coefficient of x9 in x3+x5.

Q: Find the LCM of the polynomials 1.x²³²-²³² and X²²=-2xy + y²2² 2 X HI 2.5x³y and 15x² Y

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Q: Suppose 671 + 3v2 – 203 = 0. - Let w = 301 – 6v2 – 403. Which of the following must be true?

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Q: Is it true that GCD(6,14,119) x LCM(6,143,119)= 6x143x119?

A: We know that GCD(a,b,c) × LCM(a,b,c) = axbxc

Q: Show that the system consisting of all elements of the form a+b√5, where a and b are any rational…

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