Problem 7.3 (a) Let m be a positive odd integer and a be an integer such that (a,m) = 1 and the congruence (1) x² = a (mod m) has a solution. Show that (2) (-) = 1. (b) Conversely, does (2) necessarily imply that (1) has a solution?

Q: Show that 4x2+6x+3 is a unit in Z8[x]. All that I know that I have to do is find a polynomial such…

A: Given polynomial is,

Q: 6. (a). [3-2 စာ ၅၊ = Let A 1-1-2 = 0-2 Compute A-1 ***** (b) Use part a) to fin all solutions to the…

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Q: (4) Prove in two different ways that if a is a positive r number, then (a + 1)" > a" + na"-1:

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Q: Q2: Solve the following (if possible): 1. = 5x + 11 mod 37 3

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

Q: 4. Solve each of these congruences using the modular inverses found in the previous problem. (a) 7x…

A: The given congruence equations are (a)    7x=2 (mod 26) (b)   19x=4  (mod 141)

Q: If α and β are two real numbers, we say that β is equivalent to α if there are integers a, b, c, and…

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Q: (i)Find the inverse of 5(mod9)     (  ii)Hence or otherwise solve simaltaneous equation…

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Q: Prove by induction that, for every n\geq2 n+1 (1-2)(1-¹)(1-(1-1/2) = ² 2n

A: To Prove: We have to prove by induction that for every n≥2, we have:…

Q: (a) (b) (c) (d) If P and Q are odd positive integers, we have mn ( ² ) ( ² ) = (m²). m Ⓒ) (@) = (₂)…

A: Since you have asked a question with multiple subparts , we will solve first three subparts for you.…

Q: Find all solutions to the following quadratic congruences by first making the leading coefficient 1…

A: Topic:- algebra

Q: Advanced Math Question

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Q: 5. Let f(x) = 2x² + x + 4. It is known that f(1) = 0(mod 7). (i). Find a solution to f(x) = 0(mod…

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Q: (7) The difference of two odd integers x and y is 4. Prove that x and y are relatively prime.

A: Given difference of two odd integers is 4.

Q: Solve the following quadratic congruences: (a) 3x² + 9x + 7 = 0 (mod 13) (b) 5x² + 6x + 1 = 0 (mod…

A: According to Q&A guidelines, I can answer only one question. So I answered (a).

Q: Let H = {(1), (12)(34), (13)(24), (14)(23)}. Find the left cosets of H in A4 (see Table 5.1 on page…

A: Given H = (1), (12)(34), (13)(24), (14)(23) in A4 H =4and A4 =4!2 =12 So , Number of distinct left…

Q: (2) (a) Find all of the solutions modulo 15 to 6x = 15 mod 18. Express your answer in terms of the…

A: Given, 6x≡15 mod 18. As it is known that, an≡bn mod cn ⇔ a≡b mod c. Therefore, we have 6x≡15 mod 18…

Q: 3. Compute |(1+ i)(1 – i)(1 + 3i)| .

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Q: 8 5 4. Let A -7 - (a) Compute A¨¹. (b) Use your answer to part a) to solve Ax =

A: (a) To compute the inverse of a 2×2 matrix A= [ac​bd​], we can use the formula:…

Q: O Question 10 of 12 b=0b 0 (4 3i)(4+31) D 0 (4-3i)(1) (61)(4+31) (61) (1) D (-5)(4+3) D B (-5)(4)…

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Q: Find all solutions of the congruence 3 f(x) = 5x³ - 93= 0 (mod 231).

A: The given congruence equation is .We need to find all the solutions of it.

Q: Describe geometrically the effect of the transformation T Let A = -(1) Define a transformation T by…

A: We are authorized to answer one question at a time since you have not mentioned which question you…

Q: If / (7-4) (x + 8) 12 O 4 dx Adx - A+B 4 Bdx x+8' , then A+B =

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Q: (3) (a) Find the smallest positive solution of the system of linear congruences x = 3 (mod 5) * = 7…

A: We solve this using Chinese Remainder Theorem

Q: Find all the solutions to the linear congruences  (b) 128x = 833 mod 1001 (c) 36x = 30 mod 42

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Q: Prove (1.2), (1.3), and (1.4).  

A: (a) Let x∈A∪B So, ⇔x∈A or x∈B⇔x∈B or x∈A⇔x∈B∪A So, A∪B=B∪A

Q: 3. (i). Suppose f (x) is a polynomial with integer coefficients such that f(2) = 0(mod 5) f(4) =…

A: Given that  (a) f(2)≡0(mod 5)f(4)≡0(mod 9) We have to find a polynomial such that f(x)≡0(mod 45)…

Q: (a) Male honeybees are from single parent families; they are from unfertilized eggs so only have…

A: Let Bn be the number of nth generation ancestors of a male hobeybee. Then we have   B1=B2=1. Now for…

Q: Acircle whose areais 452 cm2is cutinto two segments by a chordwhose distance from the center of the…

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Q: let f(r) = r – r' +x +x? – 1 and q = 41 Find f(x)-1 (mod q) Show all step. The answer should be…

A: Solution

Q: 1. (a) Compute A−¹ if A = /1 0 1 33 4 223 (b) Use A−¹ to solve Ax = b, if b = (1, 1, 1)ª.

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Q: Prove that Q (1 + i) belongs to Q (4-i)

A: Prove that Q (1 + i) belongs to Q (4-i)