Prove or disprove each of the following two statements. (i) For all integers x and y and natural numbers n, if xy = 0 mod n, then x = 0 mod n or y ≤ 0 mod n. (ii) For all natural numbers n > 4 and integers x, if x² = 4 mod n, then x = 2 modn.

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Q: Find an integer n such that n = 1 mod 7, n = 3 mod 8, and n = 2 mod 9.

A: We can solve this system of congruences using the Chinese Remainder Theorem.

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A: Congruence Modulo Relation on Integers: Let a, b∈ℤ and let m be a fixed positive integer.  Then a…

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A: Your questions solution below in images

Q: I want to know that the situations of "if" in the question

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Q: 4. For any integer n, show that if n³ = 1 (mod 47), then n = 1 (mod 47).

A: To show that if n3≡1(mod47), then n≡1(mod47), it can use proof by contradiction.Assume that…

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Q: Prove the assertions below: (a) If a is an odd integer, then a? = 1 (mod 8). (b) For any integer a,…

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Q: (b) Wilson's Theorem: Let p be prime. Then (p-1)!= -1(mod p). Hint: Verify for p = 2 and p = 3…

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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 n be a positive integer. We write x; for the ith digit of n. (I.e., if n = 217, then x1 = 7, x2…

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Q: Select the following statements that are true. a 0 for any integer a > 0 31000 = 1 (mod 10) 2022 1…

A: According to our guidelines we can answer only three sub-parts and rest can be reposted.

Q: (5a) Let A = {n E Z | n = 2 (mod 3)} and B = {n E Z | n = 1 (mod 2)}. Prove that if n E (A n B),…

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Q: - Show that (a + b)" = a + bn mod 2 for all integers a, b, and n with n > 0. Hint: Try looking at…

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Q: Prove for each integer n, that n2 is congruent either to 0 or to 1 modulo 4.

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Q: Need help with a&b

A: For (a), The statement is false. Counter Example: Let x = 2 and y = 3. Notice that xy = 6 ≡ 0 mod 6…

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…

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A: We reduce the congruence and then solve

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Q: do I prove this: Let a, b, y ∈ Z and d ∈ N so that b ≡ y(mod d). Then a ⋅ b ≡ a ⋅ y(mod d).

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

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Q: For any integer n, either n2 mod 4 = 0, or n² mod 4 = 1.

A: For any integer n, we prove that either n2 mod4=0,  or n2 mod4=1

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Q: Prove: for any integer x, if x mod 3 is not 0, then (x2−3x+ 1) mod 3 = 2.

A: To prove :     for any integer x, if xmod 3≢0, then x2-3x+1mod 3=2   Pre-requisite : 1. a≡bmod n…