(9) Let a, b, and m be integers with m 21 and (a,m) = 1. Prove that the congruence ax=b (mod m) has a unique solution modulo m.

Q: Given 6x + 2 ≡ 101 (mod 7), find an integer z such that 2x ≡ z (mod 7)

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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: Calculate C(12,5) by hand. As C(n,r) = n!/(n-r)!r! identify n and r, and write out every number in…

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Q: 1. Find all solutions (a) 2x + 3 = 7 (b) 2x + 3 = 7 to the following congruences.. (mod 9) (mod 20)

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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: 2r = 3 4x = 2 3x 2 (mod 5) (mod 6) (mod 7).

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Q: a) Find integers a and b such that a + b =a-b (mod 5).

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Q: Find the smallest positive integer x that solves the congruence: 7x≡5 (mod52)

A: 7x≡5 (mod52)---(1)

Q: Which of the following linear congruence has no solution? (a) 3x = 9 (mod21) (b) 6x = 4 (mod20) (c)…

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Q: Which of the following linear congruence has 1 incongruent solution? (a) 3x = 9 (mod21) (b) 14x = 12…

A: The solution is given as

Q: Which of the following linear congruence has 2 incongruent solutions? (a) 3x = 9 (mod21) (b) 14x =…

A: to solve linear congurence , ax≡b(mod m) .........(1) the first stem is to reduce the congurence  by…

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

A: Topic:- algebra

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

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

Q: Use the index system modulo 19 to find all solutions of x3≡11(mod 19) and 5x≡4(mod 19).

A: The congruences given are as follows: x3≡11mod 195x≡4mod 19 We are asked to find the value of x that…

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

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Q: 2x = 4 (mod 8) a. has no solution b. has infinite solutions c. has a unique solution modulo 8 which…

A: Here 2x≡4 (mod 8). Hence 2x=4+8k, for some integer k. Dividing by 2, x=2+4k, for some integer k.…

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

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

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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: Let p be an odd prime number and let b be an integer with p /| b. Prove that either b has two square…

A: Given:(a) Let be an odd prime number and let be an integer with does not divide . Aim:(a) Prove…

Q: 2. Solve the simultaneous congruences x = 4 (mod 5) and 2x = 3 (mod 7)

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

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Q: Solve the linear congruence 5x = 3 (mod. 11). (Show all the work.)

A: The expression, a≡b (mod c)  implies that c divides a-b, that is when c divides a-b the remainder is…

Q: 3. Find all integer solutions to the pair of congruences (if any) x = 1 (mod 2) x = 2 (mod3) x = 3…

A: To Find: Integer solutions to the pair of congruences(if any) x≡1(mod 2), x≡2(mod 3), x≡3(mod 5)…

Q: linear congruence has 2 incongruent solutions

A: b is correct. We have to find out the total number of solutions of the given linear congruence.