Problem 2 (10 points). Let m,n E N and a, b e Z. Suppose that a? = b² (mod m) and a = b3 (mod n). Prove that if (m, n) = 1, then aº = b° (mod mn). You must clearly write and justify the steps in your proof.]

Q: Let x E Z and y E Z. Prove the following statement: If y = 2 mod 4, then xy 1 mod 4.

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Q: If x² + y² = 4, then 0-12/0 00 04-4 0- - O 1. 13. de =

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Q: Calculate d

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Q: II. Give FIVE different answers for each of the following. 1. 49 = mod 10 2. 82 = mod 9 3. 100 = mod…

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Q: 1. Prove (2p+1) = 2 (mod p) when p is any prime.

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Q: Find the four solutions of x* = 1 (mod 13).

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

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Q: (d) 9r = 3 (mod 5) (e) 5x = 1 (mod 6) (f) 3x = 1 (mod 6)

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Q: Informal Exercise 20. Observe that 4| 102. Show that to find Rem(n, 4), you just need to replace n e…

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Q: 3 Let A =2E -1 0 B = 0 2 If BXT = A, then X= Select one: -3 0 a. 1 -3 O . 1 -3 O d. None of these 1)

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Q: 3.1.12 Suppose that 7x = 28 (mod 42). By Theorem 3.9, it follows that x = 4 (mod 6). (a) Check this…

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Q: Ques 3. Verify the following statements. 1. 55 - 26 = (3 - 2) (mod 4) 2. 55 - 26 = (3 - 2) (mod 4)…

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Q: Using Fermat's theorem, find 3201 mod 11.

A: Euler's Theorem states that for any natural  number n, we  have  aϕ(n)≡1(mod n) where ϕ(n) denotes…

Q: T T T F F F 11. 12. 13. If ab = 0 (mod 3), then a = 0 (mod 3) or b = 0 (mod 3). If ab = 0 (mod 6),…

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Q: 27 = x(mod 5) is true if x =

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Q: 6. Calculate the following (we'll end this homework with an easy one, of course, in the future we…

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Q: 9. Determine which of the following congruences have solutions, and find any solutions (between 0…

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Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.