(b) Which of the following congruences are soluble? If soluble, find a positive integer solution less than 79; if insoluble, explain why. (ii) 41x² = 43 mod 79.

Q: Q3 Prove that In Coth x 1

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Q: 2. Find an explicit formula for аn 3 За,-1+ 10ар-2 and ao %3D 0, ај 3D 7 An where

A: The solution is given as

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A: We will use (13y)mod35=1 to find all possible values of y by writing given congruence in expanded…

Q: ions of the congruence 15x^2 + 19

A: Given:- 15x2+19x≡5 mod 55

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Q: Prove that if gcd(a,b)=1 then gcd(a+b,ab)=1.

A: Given, gcda,b=1 Let, d=gcda+b,ab This implies, da+b and dab This gives, dba+b⇒dab+b2Since,…

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Q: Suppose x,y,z are mutually coprime positive integers that solve x2+y2=z2. a) Show why exactly one…

A: Basic properties of modular arithmetic and gcd will be used in solving this problem. The detailed…

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Q: Determine the general solution of the following Efuler equatión for x>°, g-2xy-4y= \0x using the…

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Q: Expand x?y + 3y – 2 in powers of x + 1 and y – 2. (up to third degree terms).

A: Given :- The given expansion x2y + 3y-2 in powers of x+1 and y-2.

Q: 4. Prove that 3i² + i = n(n+ 1)?

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Q: Solve the folowing equation on [0,2pi) 2sqrt2 +4cosx=0

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A: We need to find the polynomial of degree three with integer coefficient

Q: (9) Is an complex annullus Ann(2o, R1, R2) with 0 < R1 < R2 simple connected?

A: Note: As per bartleby instruction when more then one question is given only one has to be answered.…

Q: 2. The exact value of sec 57 is O2 2V3 -2 2V3 3

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Q: For which values of a does ax = 1 have a solution in Z5?

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Q: 3. Classify the following PDE, reduce it to the canonical form and then find its solution: 1 Uzz -…

A: Hyperbolic pde

Q: Solve the following Higher Order Non-homogenous DE using Methods of undetermined coefficients…

A: Given,y'' -2y' +y=3x+2A second order linear, non-homogeneous ODE has the form of   ay''+by'+cy=gxThe…

Q: 2) × if it was 2+4i and 2=3-1 3+1 2+i 2 prove that x² + 3xy + y² = 6 =

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Q: Find all the solutions r E Ziz to the equation r³ – 5x² – 7x = 0 mod 13.

A: First, solve in real number then ring down to the finite field.

Q: (a) A rectangle has area of 96 ft2 and perimeter of 40 ft. Find its dimensions. (b) Prove the…

A: 5.  (a) Given that the area of the rectangle is 96 ft2 and the perimeter is 40 ft. Let x be the…

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A: Z(t) =(2-3i)cos(2t)-i sin(2t)  Or Z= 2cos(2t)-3icos(2t) -isin(2t)  Or Z= 2cos(2t)…

Q: Find the value of m that makes: Show that the equation  xex-1=0 has a unique solution on [0,1].

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Q: a) Prove If he Given DE Is exact or non-exact Find its General solution

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Q: Consider the following IVP: x" - 4x' + 4x = 4e2t; x(0) = -1, x'(0) = -4 %3D %3D

A: We have to solve the given differential equation,(IVP).

Q: Which of the following is true regarding the linear Diophantine equation 2.x + 4y = 9? It has no…

A: We have to check

Q: 10) Which of the following shows the solution for n2 + 2n 2 24? A) 4十 -6-5-4-3-2-10 123456

A: The given inequality is n2+2n≥24

Q: The least positive solution modulo 330 for the system х3D 1(тod 5) х%3D 2(тоd 6) x = 3(mod 11) is x…

A: Chinese Remainder Theorem: If m1, m2, .., mk are pairwise relatively prime positive integers and if…

Q: 14. Find all solutions to 2sin(5x)-3V3 =-4/3. State exact values only. %3D

A: Given equation, 2sin5x-33=-43

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Q: In what case of k such that there is only one solution? 2x + ky = 10 kx + 2y = -10

A:

Q: Verify that the Diophantine equation x² + y² where d'(n) and d"(n) are the numbers of positive…

A: We will first show that product of two square numbers can be expressed as a sum of two square…

Q: HW KLX) = 2x+5 7 -1. Find ke-²¹ (x) 2. Prove by kok orklok

A: Let's find inverse function.

Q: 14.130 the ratimal 2e10s the orem to find all the real 2eroß the Dalynamial furltion, Use the zeros…

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Q: For which values of a and m does ax = 1 have a solutionin Z m?

A: To find for which values of a and m does ax=1 have a solution in Zm. Zm=0,1,2,....,m-1 when a=0,…

Q: Prove that tanodo-n//2.

A:

Q: 2. Solve for the complex mumber z in r (4+ 36) – 7 = 7x - 4zi. Show all algebra.

A: Given expression is x(4 + 3i) - 7 = 7x - 4xi

Q: 5. (8x*y® + 3x³) dx + (12x°y³ + 3xy²) dy = 0

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Q: Prove that there are no solutions in integers x and y to the equation 2x2 + 5y² = 14.

A: Two different proofs are given for the given problem as follows :

Q: The solution of e = -has the form x + i (y + 2nn) where n is an integer. What is the value of x?…

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Q: 2) × 2+4i 3+1 and 2=3-1 2+i 2 prove that x² + 3xy + y² = 6 if it was

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Q: 5n+7 Find supremum of A :|n E N 4n+12

A: Given set isℝ A = 5n+74n+12 | n∈ℕ We know that a number u is the supremum of a nonempty subset S of…

Q: 10. Find all solutions of the system of congruencies x = 2 mod 3 x = 4 mod 5 х З тod 7

A: Use Chinese remainder theorem Answer and explanation is given below

Q: 2. 2. (xij-X-) Xij-.. prove thate

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Q: 5. Find the general solutions to the nonhomogencous, lincar difference cquations. (a) x,12 + X141 –…

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Equations and Inequations

Equations and inequalities describe the relationship between two mathematical expressions.

Linear Functions

A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.