Show that if p is a prime number then every equation of the form ax ≡ b mod p with a 6≡ 0 mod p has at most one solution. Give examples to show that linear equations mod n (for n not prime) that can have zero or multiple solutions.

Q: 4. Given that 3-2i is a root of the equation z-6z³+18z²+pz+65-0, find all the roots and the real…

A: If  is a root of  then its conjugate  is also a root of  and hence  is a factor of .

Q: How many integer solutions are there to the equation with x₁ > 1, 2 > 2, x3 > 2, x4 > 0 ? X1 + x₂ +…

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Q: Prove that x113+x-1≡0(mod 113) has no solutions in x.

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Q: Identify the solution to an=2an-1+ an-2-2an-3 with ag = 3, a₁ = 6, and a2 = 0 for all integers n ≥…

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Q: quadratic equation r2 + 4x-3 0 have a solution modulo 41"

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Q: Solve the quadratic congruence, 6x²-7x+2 ≡ 0 mod 31.

A: We have to solve the above quadratic congurence.

Q: Find all solutions of the congruence 8x ≡ 4 (mod 12)

A: Step 1: Divide by the greatest common divisor (GCD)The GCD of 8 and 12 is 4. We can divide both…

Q: Find the general solution to the system of congruence equations X ≡ 1 mod 6; X ≡ 4 mod 9; X ≡ 7 mod…

A: The given problem is to find the general solution for the given system of congruence equations and…

Q: 7) Note that -2 is a square modulo 11. Indeed, 82 +2 = 6.11. Use this information, and the method of…

A: To use method of descent to construct a solution of x2+2y2=11 Now, it is known that -2 is a square…

Q: 5. Consider the equation x₁ + x₂ + x3 = 18, where 2₁, 22, and 23 are integers. How many solutions…

A: This question from number theory.

Q: 2. By using Fermat's Theorem, solve, 21x = 12(mod 17).

A: First, we simplify the equation by reducing the coefficients modulo 17. This is because in modular…

Q: Solve for x in of the following equations: x- 2x2-x+2 = 0 c) a) 3(x + 8) = 7x(x + 10) 2.

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Q: How many non-negative integer solutions (O is allowed) are there to r +y+ z = 24?

A: The given equation is, x+y+z= 24 We know that, For equation,  x1+x2+......xr=n  There are n+r-1Cr-1…

Q: Solve the linear congruence 2370x ≡ 69 (mod 2517) or say why no solutions exist

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Q: How many nonnegative integer solutions are there to the equation Xị + x2 + x3 + x4 + x5 = 40 if we…

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Q: Solve the system 3x + 2y = 5 2x + 7y= 9 mod 17 mod 17

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Q: How many non-negative integer solutions are there to the equation rị + *2 + x3 = 21

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Q: Solve the congruence17x−3 = 4 (mod 26).

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Q: 5. Find all integer solutions to 6s+ 14t+21u = 1.

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Q: 2.23 Find all four solutions to the equation x²10 (mod 35).

A: The given equation is: x2-1≡0 mod 35 We have to find all four solutions to the given equation.

Q: Find a general solution of ди ди 2- + 3 2u ar

A: Given the pde

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

A: Topic:- algebra

Q: 3) Suppose that f mx² + ax + bdx = m, a, & b are constants and C is an arbitrary constant. For each…

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Q: Prove that there are no integer solutions to the equation x² = 4y + 3

A: To prove that there are no integer solutions to the equation…

Q: Let A = 2 21 -1 1 1 0 3 3 ---B 3 Calculate A¹, and use A to solve Ax

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Q: For any numbers a andb and an even natural number n, show that the equation x" + ax + b = 0, x in R,…

A: Here, to prove this result we have to use Rolles theorem.

Q: Solve the congruence3x−7 = 2 (mod 16). Your solution should be positive and reduced modulo16.

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Q: Prove that if a > 0, and b is an arbitrary real number then x³ + ax + b= 0 has exactly ne real…

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Q: The set of exhaustive values of m for which both roots of the equation x2 – 2mx + m2 – 1 = 0 are…

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Q: 59. Let a, c +0. Show that the roots of ax? + bx +c = 0 cx? + bx + a =0 and are reciprocals of each…

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Q: Find all the non-congruent solutions to  4x ≡ 6(mod 21)  express solutions using the stand…

A: Introduction: A linear congruence is generalized as ax≡bmod n. The greatest common divisor of a and…

Q: 2z Solution for 3* 9 ²² * 27 ²-1 a) -1/2 b) 1/2 c) 1/7 d)-1/7 e) 1/9

A: I have appended the formula used in the solution procedure

Q: . Find all the solutions of 3x² + 2x + 1 8 (mod 63).

A: Given a quadratic congruence in the form:Ax2+Bx+C=0(mod n)multiply the above equation by…

Q: Find all the solutions of 15z = 27 (mod 18).

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Q: 3. Let a be a primitive 10th root of unity. Which of the following is false? A. a5 = 1 B. a³ = a¹3…

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Q: Solve the quadratic congruence x² = 35 (mod 817).

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Q: How many integer solutions are there to the equation x+y+z=10 if x, y, and z are all bigger than -2?

A: Given, x+y+z=10 x,y,z>-2

Q: Find the number of 5-tuples (a, b, c, d, e) of positive odd integers that are solutions to the…

A: The idea is to transform this into a known counting problem

Q: 9. Find all the solutions of 3x² + 2x + 1 8 (mod 63).

A: