3. For the following statements, either prove that they are true or provide a counterexample: (a) Let a, b, m, n ∈ Z such that m, n > 1 and n | m. If a ≡ b (mod m), then a ≡ b (mod n) (b) Let a, b, c, m ∈ Z such that m > 1. If ac ≡ bc (mod m), then a ≡ b (mod m) (c) Let a, b, c, d, m ∈ Z such that c, d ≥ 1 and m > 1. If a ≡ b (mod m) and c ≡ d (mod m), then a^c ≡ b^d (mod m)

Q: 6. Let m, n EN with m, n > 2, and let a, b E Z. Prove that a = b (mod mn) implies a = b (mod m) and…

A: (a) a = b (mod mn)       => there exists some integer k such that                    a = b +…

Q: T F 5. To prove the statement: If ab =0 then a = 0 or b = 0, you may assume ab = 0 and a # 0 and…

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Q: Write a proof for the following statement: (Use direct, contrapositive, or proof by contradiction.)…

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Q: Let a, b, n be positive integers. Prove that if a = b (mod n), then GCD(a, n) = GCD(b,n).

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Q: Are the following statements true or false? Either prove the statement is true or provide a…

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Q: How do I prove this:  Let d ∈ N and let a, b ∈ Z. Prove or give a counterexample: If a 6≡ 0 (mod d)…

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Q: Find n ∈ {1, 2, 3, . . . , 28} that satisfies 26! ≡ n (mod 29).

A: Given ,              26! ≡ n mod(29)  Then we have to find the  value of n∈  1 , 2 , 3 , .........,…

Q: Let A = {r € Z | r mod 15 = 5} and B = {x € Z| r mod 3 = 2}. Give an outline of a proof that A C B,…

A: Suppose we have two non-empty sets A and B . If x∈A then x∈B for all x∈A then A⊆B . If x∈B then x∈A…

Q: 1. Let a, b e Z. Prove that if a = 0 (mod 5) and b = 2 (mod 5), then a² + b² = 4 (mod 5).

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Q: Prove that for all integers x, if x is not divisible by 7, then x³ = 1 mod 7 or x³ = −1 mod 7.

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Q: Let n e Z. Prove that n² + 4 is divisible by 8 if and only if 3n + 6 is divisible by 12. (Hint: the…

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Q: 4. Let p be a prime number and suppose that p = 3 (mod 4). Let A be any integer. Prove that exactly…

A: To prove that exactly one of the congruences X2≡A(modp) and X2≡−A(modp) has a solution, where p is a…

Q: Let m e N and a E Z. Suppose that 3|(m) and that a is a primitive root (mod m). Prove that a is not…

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Q: prove, for all integers x, if x mod 3 does not equal 0, then 3 | (x2 +2)

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Q: 5. Let a, b e Z and n E Z*. Prove that if a = b(mod n), then gcd(a, n) = gcd(b, n). (Hint: use the…

A: The greatest common divisor (gcd) of two or more numbers is the greatest common factor number that…

Q: Let n E Z. Prove that n2 # 2 (mod 3).

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Q: Let a, b, c, m be integers with m > 2. Prove that if ac ≡ bc mod m and d = gcd(m, c) , then a ≡ b…

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Q: Proficiency #7. [Counterexamples] For each of the following two statements, give a counterexample to…

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Q: 1. Let m₁, m₂ € N and suppose that a, b = Z satisfy a = b (mod m₁) and a = b (mod m₂). Prove that,…

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Q: (12) Let a,b, m, nE Z. Assune if greater than ave Prove mln and a = b (mod n) that then a = b (mod…

A: Congruence equations

Q: Let a, b e Z and n E Z*. Prove that if a = b(mod n), then gcd(a, n) = gcd(b, n).

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Q: Prove or disprove each of the following two statements. (i) For all integers x and y and natural…

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Q: 2. Let A = {n? +n € Z:n e Z} and B = {r e Z : r³ – 3x³ + 2x = %3D O mod 5}. Prove or disprove: A C…

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Q: Exercise 18.3.13. (a) Verify Euler's theorem for n = 15 and a = 4. (b) Verify Euler's theorem for n…

A: First we calculate phi(15) and phi(22). Then we evaluate lower powers of 4^n and 3^n and see if we…

Q: Prove the following statement or disprove it by giving a counterexample: "For all positive integers…

A: The given statement, "For all positive integer n and for all integers a and b, if a2≡b2modn then…

Q: III. Determine the value of a such that: 10. a = 11 (mod 21) and 90 s a s 110 11. a = 19 (mod 29)…

A: The given questions are related to congruences. The solution is explained below.

Q: I know that using a corollary that the congruence x2 ≡ 0 (mod p) that the only solutions are + and -…

A: As per the question we have the following congruence relation : n ≡ 2(mod 4) And we have to prove…

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: Prove that if a1, a2, . . a, are n > 2 integers such that a; = 1 (mod 3) for every integer i (1 < i<…

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Q: Find an integer y, 0 ≤ y ≤ 18, that satisfies the following congruence: 13y + 9 ≡ 10 (mod 19). You…

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Q: 12. Prove that if gcd(a, n) b, then ax = b (mod n) has no solutions.

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