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

Q: (1) Determine the number Sm of solutions to the equation xyz = 1 (mod m) for each positive integer…

A: Given that:To find:The number  of solutions to the given equation for each positive integer…

Q: Find all the solutions of the congruence equation x6-2x5= 35 (mod 6125).

A: Consider the given congruence equation. x6-2x5≡35mod 6125. Now, the prime factorization of 6125 is:…

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

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Q: Find all solutions for x, up to congruence.  x ≡ 2^63 (mod 61).

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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: Find all solutions of the congruence x2 ≡ 29 (mod 35).

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Q: Show that x2 = Y mod p has

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Q: 2. Let a be a positive integer. Explain why the linear congruence ax = b (mod a + 1) has a solution…

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Q: Let a and b be positive integers and let b= qa +r, where 0 ≤r <a. Prove that (a) 2b - 1 = 2" - 1…

A: Let  and  be two positive integers.Let  where (a) We  have to prove that      Now, observe that…

Q: Find all four solutions to x2 =6 (mod 19 x×23) with 1<x< 437.

A: Solution:-

Q: Find all solutions in {0, 1,2, ..., 12} to x26 + 15x13 = 12 mod 13.

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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: a) Find all solutions to the congruence 10r = 5 mod 15

A: Given,

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

A: Topic:- algebra

Q: Find all solutions for x, up to congruence. (b) x ≡ 11 mod 142 and x ≡ 25 mod 86.

A: Given: (b) x≡11 mod 142 and x≡25 mod 86 To find: All solutions for x, up to convergence.

Q: Find the solutions of the congruence 15x2 + 19x ≡ 5 (mod 16).

A: Congruence modulo relation on integers:Let a,b∈Z and let m be a fixed positive integer. Then we say…

Q: (2x + 3 ) ≡ 8 mod 12

A: (2x + 3 ) ≡ 8 mod 12 x + 3 + 9 ≡ 8 +9 mod 12  2x + 12  ≡ 12 +5  mod 12

Q: 2. Find all solutions of the linear congruence 12x = 9 (mod 15).

A: P

Q: Find the solutions of the congruence 15x2 + 19x ≡ 5 (mod 65)

A: The given equation is: 15x2+19x≡5(mod 65)

Q: Use Hensel's Lemma to find a solution to the following congruence: a +3 x + 3 = 0 mod 54.

A: Let f(x) be a polynomial , p is a prime number . If x=a is a solution of congruence equation…

Q: Find all whole number solutions of the congruence equation. (2x + 1) ≡ 5 mod 4

A: Consider the given congruence equation. (2x + 1) ≡ 5 mod 4

Q: Find all integer values of x for which 3x = 4 (mod 8).

A: given congruence equation 3x≡4(mod 8)

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

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Q: Find all the solutions to the linear congruences  (b) 128x = 833 mod 1001 (c) 36x = 30 mod 42

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Q: ind all solutions of the linear congruence: 9x = 5 mod (14)

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Q: Give all non-negative solutions for 10r =-17 mod 4.

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Q: Find all solutions of the congruence x2 ≡ 16 (mod 105).

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Q: Does x = a (mod p) always have a solution for every value of a, whenever p is prime?

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Q: 1. For each solution. congruence determine whether there is a J X² = -1 (mod 397) (397 is prime) CC…

A: We shall be using the result regarding the value of legendre symbol

Q: Find all solutions to the congruence 10x≡5 mod 15 (b) Prove that 7 divides 32n+1+ 2n+2for any…

A: In  (a) we use the extended Euclidean algorithm to solve the problem. In (b) we use the mathematical…

Q: Determine whether the given congruence(s) is/are solvable and find all incongruent solutions if they…

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Q: Find all the solutions 2 The congruence x² = 58 mod 77

A: The congruence is x2≡58mod 77. To illustrate the solution to given congruence. Here 77 can be…