Find all possible congruence classes x mod 6 such that 2x + 3 = −1 (mod 6).

Q: The statement a ≡ b (mod n) is called a congruence.

A: Given that  The statement a ≡ b (mod n) is called a congruence.

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

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Q: ​Q3// Use Chinese theorem to solve the following congruence: X ≡ 2 mod 3 X ≡ 1 mod 5 X ≡ 2 mod 4

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Q: Find all x (mod 1999) such that x³ = 2 (mod 1999).

A: We use primitive roots to solve this problem.

Q: (10) Find all solutions of the congruence 2 = 22 (mod 53). C

A: We shall be using primitive roots to solve this congruence.

Q: Express each nonnegative integer a less than 15 as a pair (a mod 3, a mod 5).

A: The values of a could be, 0,1,2,3,4,5,6,7,8,9,10,11,12,13,14.

Q: Q// Let m = 5, 1- Write the Addition table (Cayley's Table). 2- Write the congruence classes of…

A: For m=5 , our set is Z5 .

Q: . Determine whether or not x³ + 2x + 1 and x4 + 3x - 2 are congruent mod (x² + 2x + 2) in Q[x].

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Q: Prove that 2 is not a primitive root modulo 31.

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

A: we solve the modular equation

Q: Find the order of all element of Z8 under addition mod . Please brieflt explain and show all the…

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Q: Prove that a^561 ≡ a mod 561.

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Q: (a) Prove that 2 and 3. are both primitive roots modulo 5.

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Q: (7) (Z/50Z)*. Which of the other elements of (Z/50Z)* are also primitve roots modulo 50? Find a…

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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: Use Euclid's Algorithm to find a multiplicative inverse of 23 mod 75. Once found, solve the linear…

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Q: 1. For each solution: determine weather there is a a congruence a x² = -1 (mod 397) b.) x²+8x+14=0…

A: 1a) To determine whether the congruence equation x2 ≡ -1 (mod 397) has a solution, we need to check…

Q: Prove that if gcd(a, m)=1 and gcd(a - 1, m) =1, then 1+a+a²+...+ a² + a(m)−1 = 0 (mod m).

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

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Q: Prove that the number of solutions to the congruence xk ≡ 1 mod p is gcd(k, p − 1)

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Q: (2) Compute the order of a modulo 11 for a = 1,2,..., 10. Find all primitive roots modulo 11.

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Q: 2 has order m modulo (2m - 1).

A: To prove that 2 has order m modulo , we need to show two things:For any positive integer

Q: SImalest posiuiy OI the n) is order of a modulo n. Find the additive orders of each of the following…

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Q: 8) Prove that 2 is a primitive root modulo 53.

A: We have to Prove that 2 is a primitive root modulo 53.

Q: Show that there are exactly (p−1)/2 nonzero squares mod p.

A: To prove that there are exactly p-12 nonzero squares mod p.

Q: for a fixed n>1 Show that all the solvable congruences x2 ≡ a ( mod n ) have the same number of…

A: To Proof: For a fixed n>1 , all the solvable congruences x2 ≡ a ( mod n ) have the same number of…

Q: Let a and b be positive integers with (a, b) = 1. Show that if g is a primitive root (mod ab), then…

A: We have given (a,b)=1 where a, b both are positive integer. We have to show if g is primitive root…

Q: For which positive integers a is the congru

A: Given,               ax4≡2 (mod 13)

Q: Consider the congruence 9x   21 (mod 33). (a) Does 9 -1(mod 33) exist? Explain why or why not.

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Q: Decide whether the following binomial congruence has any solutions x2" = 10 mod 103. Use modular…

A: We have to decide whether the following binomial congruence has any solutions or not.…

Q: Prove that if x2 is congruent to y2 (mod n) and x is not congruent +or - y (mod n)  then gcd(x + y,…

A: Given: x2≡y2modn x≢±ymodn

Q: (a) Use cases based on congruence modulo 3 and properties of congruence to prove that for each…

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Q: Are the polynomials x³+2x +2 and x4 + 3x -2 equal to mod(x² + 2x + 2) in Q[x]?

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Q: B- What is the multiplicative inverse of x¹5 in Z₂[x] (mod x² + x + 1)?

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Q: Which of the following satisfies the congruence 3x O O 2 T - 2 (mod 5)?

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Q: Find an integer x satisfying the two congruences: x = 1(mod 3) 2x = 0(mod 6), or show that no such…

A: The given congruences are                                 x≡1 (mod 3)2x≡0 (mod 6)