Prove the following statements using either direct or contrapositive proof. 1. Let x,y eZ. If x+y is even, then x and y have the same parity. 2. Let a,b,ceZ and neN. If a b(mod n) and a c(mod n), then C= b(mod n).

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: Prove the following statement or disprove by giving a counterexample "for all positive intergers n…

A: We have to prove or disprove the following statement For all positive integer n and for all integer…

Q: I. Solve for the smallest value of x in the following linear congruences. 1. x = 3 mod 6 and x = 4…

A: To solve given linear congruences

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: 1. Use Euler's theorem to establish the following: (a) For any integer a, a [Hint: 1729 = 7. 13 ·…

A: As per the question we will use the following Euler's theorem : -  aφ(n) ≡  1 (mod n) Where φ(n) is…

Q: If x = y (mod n) and u = v (mod n) then which of the following is false. (a) x+ u = y +v (mod n) (b)…

A: Given that x≡ymod n and u≡v mod n. By definition, x≡ymod n implies x-y is divisible by n. Then let,…

Q: 2. Prove directly if the statements that are true, give counterexamples to disprove those that are…

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Q: mod

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Q: If a = b (mod n) and a = c (mod n) then which of the following is true. (a) b3 c (тod n) (b) b2 =…

A: Given below the detailed solution

Q: Solve the following systems of congruences. a. x = 2 (mod 5) x = 3 (mod 8) b. x = 4 (mod 5) x = 2…

A: Given system of congruence are a) x≡2(mod5)    x≡3(mod8)b) x≡4(mod5)    x≡2(mod3)

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

A: The given problem is to prove the following identities, we have to use basic identities of modular…

Q: 2. Prove directly if the statements that are true, give counterexamples to disprove those that are…

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Q: Is the following proposition true or false? Justify your conclusion. For each a e Z, if a # 0 (mod…

A: The given proposition is as follows. For each a∈ℤ, if a≢0mod 3, then a2≡1mod 3. The given…

Q: 3. Use the above theorem to prove that if p E N is a prime number such that p = 1 mod 4, then p is…

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Q: Let a, b, c and meZ, and m 2 2 1. Prove: if a =b (mod m), then GCF (a, m) = GCF (b, m) %3D m 2.…

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Q: 1. Prove each of the following assertions: (a) If a = b (mod n) and m | n, then a = b (mod m). (b)…

A: Given below the detailed solution

Q: Q3. Prove that if ax = ay (mod m) , then x = y (mod m/d) where d = gcd( m, a).

A: According to our policy at Bartleby, I can solve only one question. I am solving Question 3. Let's…

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: 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: Consider: for any integers a and b, if (a3 + b3) mod 3 does not equal 0, then (a + b) mod 3 does not…

A: The contrapositive of A implies B is not B implies not A. We use this and solve this problem.…

Q: Let C={x €Z: x =7 mod 9} and let D={x € Z,x = 1 mod 3). a) List five elements of C and five elements…

A: Since you have posted a question with multiple sub-parts, we will solve first three subparts for…

Q: For which p= 3 mod 4 is i a square in F»[i]? For which p = 3 mod 4 is i the square of an element in…

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Q: 3 23) Give a counter example to disprove the foilowing Property ab = ac (modn) →6=C (modn) yb=C…

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Q: Give an example to show that a? = b? (mod n) need not imply that a = b mod n). fa=b(mod n) prove…

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Q: Q1 Prove or disprove i. If ac = bc(mod m), then a = b(mod m).

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Q: 7. Let n E Z. Prove that 9n3 6 (mod 27).

A: The objective is to prove the above statement.

Q: Let c, d e Z and m e N. For each statement decide whether it is necessarily true, or whether it can…

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Q: The congruence x² = 2(mod7) is not solvable: O False O True

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Q: Let S := {1< a<p - 1: () = 1} . Ifp = 1 mod 4, show that Σ p(p – 1) 4 rES

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Q: 20. Are the following statements true or false? Either prove the statement is true or provide a…

A: PART A For the integers a and b, a.b=0(mod 6) This implies that either a=0(mod6) or b=0(mod6). So,…

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: 2) If a? = b² (mod n), then a = b (mod n) a) True b) False O a) True O b) False

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Q: Prove directly if the statements that are true, give counterexamples to disprove those that are…

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Q: Let c, d e Z and m E N. For each statement decide whether it is necessarily true, or whether it can…

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Q: If a = b (mod n) and b = c (mod n) then which of the following is true. (a) a = c (mod n) (b) a? =…

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Q: 4.14. Let a, b, n E Z, where n > 2. Prove that if a = b (mod n), then a2 = b (mod n). 4.15. Let a,…

A: The detailed solution is as follows below:

Q: If a = b (mod n) and a = c (mod n) then which of the following is true. (a) b = c (mod n) (b) b2 =…

A: We know that the property of congruence

Q: Suppose that a and b are integers, a = 4 (mod 13), and b = 9 (mod 13). Find the integer c with 0 < c…

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

A: Given if a≡bmodn and c≡dmodn We have to show that ac≡bdmodn. Now, a≡bmodn…

Q: Let n E Z. Prove that if 5n2 +3 =0 (mod 8), then 4 n.

A: In this question  we will be using contradiction concept so assume it is true statement then we…

Q: Deduce the following statements: i) an odd prime number p is the sum of two squares of integers if…

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Q: Given that a, b, c,m are integers with m > 0. If a = b (mod m) then prove that ас3 bе (mod m)

A: we have to use the only definition of congruence and the definition of divisibility

Q: 1. Prove each of the following assertions: (c) If a = b (mod n) and the integers a, b, n are all…

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Q: 1. Prove each of the following assertions: (a) If a = b (mod n) and m | n, then a = b (mod m).

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Q: (a) Consider the following statement: For all n € N, x(x − 1) = 0 mod n Which of "if", "only if",…

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Q: Consider the relations E9 and E12 on the set of integers defined as follows • aE9b precisely when…

A: Given relations aE9b precisely when a≡bmod 9 aE12b precisely when a≡bmod 12 where relation R=E9oE12.

Q: 1 Prove each of the following assertions: (a If a = b (mod n) and m |n, then a = b (mod m). (6) If a…

A: We know that  a ≡ b (mod n) is equivalent to n | (b-a)

Q: Q3. Prove that if kx = ky (mod m) , then x = y (mod m/d) where d = gcd( m, k).

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Q: Which of the following is not correct ? de = 1 (mod (p – 1) (q – 1)) Numerical value of NO is 1314…

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