5. Using the Quadratic Reciprocity Law, prove that P 1 if p=1 or 11 (mod 12), 1 if p = 5 or 7 (mod 12) -- ifp=5 for each odd prime p.

Q: What is the number of solutions to the congruence in Z125? x³ + x² + 3 = 0 (mod 125)

A: As per the question we have to solve the following polynomial congruence equation : x3 + x2 + 3 ≡ 0…

Q: (3). (a). Find all solutions to the congruence 6x = 9 (mod 45)

A: To Determine :-  All solutions to the congruence : 6x ≡ 9   mod 45 .

Q: 2. What is the least residue of 610 C 512 10.4512 (mad 1112

A: Euler's theorem is used

Q: Solve linear congruence 45x ≡ 60 (mod 75)

A:

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: Find the smallest positive integer x that solves the congruence: 4x7 (mod 23)

A: we solve the modular equation

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

A:

Q: (26) Are there natural numbers n, m such that 20" = 50m? Justify your answer with a proof or a…

A: The idea is to use a prime factorization of the numbers 20 and 50 and then use this to solve the…

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

A:

Q: Using the Law of Quadratic Reciprocity, prove that for an odd prime p #3 (²) 1 if p = ±1 (mod 12) -1…

A:

Q: x43 = 47 (mod 713)

A: Step:-1 By Chinese remainder Theorem (CRT),  x43 ≡ 47 mod 713 if and only if x43 ≡47 mod 23 ⇒x43 ≡1…

Q: Solve the system of congruences 2x ≡ 1 (mod 5) 3x ≡ 9 (mod 6) 4x ≡ 1 (mod 7) 5x ≡ 9 (mod 11) Hint:…

A: We can solve this system of congruences using the Chinese Remainder Theorem. First, let's write each…

Q: Solve the congruence 11x = 45 (mod 97) where x is an integer in the interval [0,97).

A:

Q: 3. Let x be an integer. Use the previous result to prove the following corollaries. (a) If x divides…

A: Divisibility: If a and b are integers, we say that a divides b, and write a|b, if there is an…

Q: Solve the givnen congruence x2 ≡ 9 (mod 600).

A: The given problem is to solve the given congruence equation. x2=9 (mod 600).

Q: 4. Use Chinese remainder theorem to compute 435789 (mod 66).

A:

Q: 5 points) Use a proof of the contrapositive to prove the following statem For all integers a, b, if…

A: The contrapositive of a statements  " If p then q  '' is    " If not q then not p "

Q: 17. Evaluate five non-obvious Legendre symbols (4) for p = 47 using quadratic reciprocity.

A:

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

A: Topic:- algebra

Q: solve the congruence of  6x=4 mod 10  aolve the conguence of 341x=2941 mod 9

A:

Q: Show that if p > 1 and p divides (p − 1)! + 1, then p is prime.

A: Proof by contradiction: Assume p is not prime ⇒p is composite

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: Suppose that p and q = 4p+ 25 are both prime. Show that p is a quadratic residue mod q.

A: Given p and q are both prime. and q=4p+25 To show that p is a quadratic residue mod q.

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

A:

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

A:

Q: . (a) How many quadratic residues are in the residue set Z19? (b) Determine if 4 and 9 are quadratic…

A: Introduction: Quadratic residue is related to congruent modulo theory. Since, it deals with…

Q: Solve by using the Chinese remainder theorem for the congruence X 2 (mod 7) X 4 (mod 5)

A: Given that This can be written as,

Q: 96. Solve the congruence 13 – 29x2 + 35x + 38 = 0 (mod 195).

A:

Q: 5 is a quadratic non- residue mod 17

A: The quadratic residues mod17 are { 1,2,4,8,9,13,15,16}

Q: What is the number of solutions to the congruence in Z125? x³ + x² + 3 = 0 (mod 125)

A: In the question we are given the following polynomial congruence equation : x3 + x2 + 3 ≡ 0 (mod…

Q: Solve the linear congruence 7а + 3у %3D 10 (тod16).

A: Consider the linear congruence 7x + 3y ≡ 10 mod 16.Since gcd (7, 3) = 1 we know at least one…

Q: Work the modular arithmetic problem. (12 9)(mod 3). (12 9)(mod 3) =(mod 3)

A: The given modular arithmetic problem is -  (12·9)(mod 3) We have to solve the above problem.

Q: Why does the linear congruence 25x ≡ 30 (mod 75) have no solutions?

A: To show that the linear congruence 25x≡30mod 75 have no solutions.

Q: 5. Use the Chinese Remainder Theorem to solve the simultaneous congruences: x = -1 (mod 7), x = 3…

A:

Q: 4x + 8y + + 7y х+ 2x Z 3z Зу + 2z 2 -14 3

A:

Q: Show that 2 (p - 3)! + 1 = 0 (mod p).

A: