Show that if a = b mod m and b = c mod m, then we have a = c mod m. Use the definition of congruence mod m to do this.

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 if a = b (modn) and m/n, then a = b (modm).

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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: Let a, be Z and n E N. For each statement decide whether it is necessarily true, or whether it can…

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Q: Find all solutions for x, up to congruence. x ≡ 1 mod 4 and x ≡ 7 mod 13.

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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: 2. If a = 1 (mod p), then a=1 (mod p). Select one: True False

A: If , then . The truthfulness of the given statement can be proved by taking an arbitrary value for…

Q: (b) Prove that if a = b (mod m) and if b = c (mod m), then a = c (mod m).

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

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Q: Solve the system of congruences given below. x ≡ 1 (mod 7) 3x ≡ 1 (mod 8) 2x ≡ 5 (mod 9)

A: Given system of congruences: To find:Solve the system of congruences. Chinese Remainder Theorem:Step…

Q: On the extended complex plane, ∞ is the point at the infinity. Find the limit if it exists, or…

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Q: y(5-cos xey-y²) 2.xy +1 sin x dy 5-c -cos xe sin x d. xy + In y +ein x = 5x [Answer: dx or 2xy +- y

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Q: Suppose x F and you discover that x¹ = 1 mod p and x² = 1 mod p. Please prove that xgcd(m,n) = 1 mod…

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Q: Let n be a positive integer and let ged(b, n) = 1. Show that bx by (mod n) if and only if x=y(mod…

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Q: Exercise 1. Suppose that a and b are integers, a = 3(mod11) and b = 8 (mod11) and c = a3 - b3. Then…

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

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Q: solve the following system of linear congruences x ≡ 2(mod 14) x ≡ 16(mod 21) x ≡ 10(mod 30)

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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: #5. Suppose that a = 2 (mod 113), and b = 5 (mod 113) What is (a b + 56b + 155) mod 113?

A: given a≡2(mod113),b=5(mod113)a3≡23(mod113)a3b≡235(mod113)56b≡280(mod113)56b≡54(mod113)155≡15mod113)

Q: Prove that if a = b (mod m) and if c = d (mod m), then ac = bd (mod m).

A: a≡b   (mod m)⇒ m|(a−b)That is there exist an integer k such that a−b=mk

Q: he linear congruence 132x = 76 (mod 100) has a solution. elect one: D True D False

A: I will use the property of linear congruence relation.

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: Recall that a = b( mod n) if a = b+ kn for some integer k. Use this to prove that if a = b ( mod n)…

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Q: 2. Prove that for any m = 1 mod 11 that there are no integer solutions of m = 2r" - 9y.

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Q: If a =b (mod m) then a" = b" (modm) Vn e Z*.

A: Given a≡bmod m. we have to show that an≡bnmod m. we wil prove this by result by using method of…

Q: Ha = b (mod n) and gcd (a,n)= d, then gcd (b,n) %3D

A: Given a≡b(mod n). This can be written as a-b≡0(mod n). Implies a-b=kn for some natural number k.…

Q: Determine all solutions x ∈ Z to the triple x ≡ 5 (mod 6), x ≡ 7 (mod 10) and x ≡ 2 (mod 15) of…

A: We reduce the congruence and then solve

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

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Q: Show that: if a = b(mod n) and C = d\modn), then a +c =b+d\mod n)!

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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: Find all possible positive values of m: 47 = m mod (5)

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Q: Consider the congruence a² + x + 1 = 0 mod 21. Solutions to the congruence above must also be…

A: The given congruence equation is .We will solve this congruence relation using the Chinese remainder…

Q: 4. A linear code is given by the encoding function y = ax+b (mod 26) and the decoding function x =…

A: 4.(a) Given a = 11, b = 2, the encoding function is given by         y ≡ 11x + 2 (mod 26)…