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

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: 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.

A:

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: 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 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: Prove that the order of 3 modulo 19 is a divisor of 18.

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Q: True or False : a – c ≡ b – c (mod n) ⟺ a ≡ b (mod n)

A: We have to solve given problem:

Q: Test2

A: We know that2198=2128×264×24×22Also we have22≡4(mod 397)24≡16(mod 397)28≡256(mod…

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

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

A:

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: Is 3 a primitive root of 43? Use your answer to determine all positive integers less than 43 whose…

A: To determine whether 3 is a primitive root of 43, we need to check if it generates all the positive…

Q: Let (a,b)=1 with a = c (mod b) and bª = c (mod a) for some integer c. Prove that ab+(b). +ba+(a) = c…

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Q: Show that if a ≡ b (mod m) and c ≡ d (mod m), then a + c ≡ b + d (mod m).

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Q: 2. Encrypt the message STOP POLLUTION by translat- ing the letters into numbers, applying the given…

A: let's encrypt the message "STOP POLLUTION" using the given encryption functions one by one.For the…

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: 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.

A:

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: Let a and b be relatively prime positive integers. Show that a(b) + b(a) = 1(mod ab).

A: Solution:

Q: Prove that 13n13 + 11n" is always divisible by 24.

A: Consider the given expression.

Q: Solve the system of congruences r = 1 mod 10 = 2 mod 7

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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: If a+c=b+c (mod m) then a =b (mod m) Select one: True False

A:

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)

Q: (b) Give an example of a composite integer n such that the congruence x =1 (mod n) pas more than two…

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Q: Use Chinese theorem to find all positive integers that makes this congruence true:        x = 2 mod…

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Q: Find all integers sx such that 2x = 5 (mod 7).

A:

Q: If gcd(m,n) = 1 is a given condition, how can you prove that the congruences x ≡ a (mod m) and x ≡ b…

A: The congruences have a solution when .

Q: Use the congruence c= (3p + 1) mod 26 to encode the message COLOR MONITOR.

A: