15. Suppose x € Z. If x³ - 1 is even, then x is odd. 17. If n is odd, then 8|(n²-1). 19. Let a,b € Z and neN. If a = b (mod n) and a = c (mod n), then c = b (mod n).

Q: 9. Consider modulo 11. a. Find 35 mod(11). b. Find a negative integer x such that x mod(11) is…

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Q: Please give detailed steps!

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Q: Given |a,b,c,n €Z, show that, if ac bc (mod n), a = b (mod), d= mdc(c, n). then

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

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Q: 4.75. Let n, m E Z. Prove that if n =1 (mod 2) and m = 3 (mod 4), then n² + m = 0 (mod 4).

A: The detailed solution is as follows below:

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: 1. Let a, b, c, r, s, and t be integers with r>s>t>1. If a = b (mod r), a = c (mod s), and gcd(r,s)…

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Q: The set of integer number is cinbounded set

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Q: 4. Suppose p = 2 (mod 5) and p is a prime number. Show that if 5 = 1 (mod p), then x = 1 (mod p).

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Q: The equivalent of Z sub 5 is set of non-negative integers less than 5. What will be the value of 2…

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Q: For D = {1, 2, 3, 4, ..., 999}, find a counterexample to the claim, "For all n E D, if n mod 4 = 2…

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Q: 1.Let a, b, and c be integers, where a≠0. If a | b and a | c, then a | (b + c)            True or…

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

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: 5- If a = b mod(n), then ac = bc mod(n). 6- If a b mod (n), then ak = b*mod(n) for every positive…

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Q: 7. Show that if n is odd, then p(4n) - 2 q(n). =

A: 7. Show that if n  is odd,then phi(4n)=2phi(n)

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: Q6. Using the definition of x = y mod N, justify the rule x = x' mod N, y = y' mod N = x x x' = y x…

A: Using the definition of modulo N.

Q: Let a, b, and c be natural numbers. Prove a ∙ (b + c) = a ∙b + a ∙ c.

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Q: 5. Let p = 61. Find all incongruent elements modulo 61, that have order: (а) 5 (b) 10 (с) 60

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Q: The order of the integer 4 modulo 19 is: O 4 O None of the choices O 6

A: To find The order of the integer 4 modulo 19.

Q: 15. Give examples of the following. Justify your answers.: (b) A set X such that |P(X)| = 8. (c)…

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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: . If x is an even integer, then x² is even. - If x is an odd integer, then x³ is odd. . If a is an…

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Q: Find the smallest positive integer for which x mod 3 = 2 and a mod 4 = 3 What is the next smallest…

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Q: 1. Let p be prime and let a 0 (mod p). Let b = a 2 (mod p). Show that ab = 1 (mod p). 2. Use the…

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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…

A: We need to find the remainder when n is divided by 4 .

Q: 5.4 For which primes p = 2, 3, 5, 7, 11, 13, 17, 19 is it true that a³ ≈ 6³ implies a ≈ 6 modulo p?

A: We make use of a well known result to solve the problem. The result is provided in step 2.

Q: 1. Using set notation, determine each solution set. . х %3D 1 тоd4&x 3D 1 тоd 5

A: Here questions 1,2,3 are unrelated since anyone can be solve without any help of another. So by…

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: 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: - 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: 38. It is known that 14 is a primitive root for the prime p = 30000001. Let 14⁹000000 (mod p). (The…

A: A primitive root mod p is a number a such that a has order p-1. This means p-1 is the least positive…

Q: Proposition 7. Let a, b, c, d E Z and n be a positive integer. If a = b (mod n) andc = d (mod n),…

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Q: 1. Let a, b, c E Z and n E N. Assume a = b (mod n) and b = c (mod n). Prove that a = c (mod n).

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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: 3. Suppose p = 2 (mod 5) and p is a prime number. Show that if x5 = 1 (mod p), then x = 1 (mod p).

A: We shall use basic modular arithmetic for solving this problem

Q: 4. Please

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