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: . If 4x - 3 = 19 (mod 31), then the value of x (mod 31) is O 26 О 18 D 21 D 30

A: Arithmetic modulo

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: 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…

Q: (a) 55 mod 7. (b) –101 div 3.

A: (a) 55 mod 7  As 55 = 7•7+6 So that  55 mod 7 = 6 mod 7

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: 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: x and y are two positive integers such that y > 5x. Select the expressions that are equal to gcd(x,…

A: The given details: x and y are two positive integers such that y>5x Assume y=5x. Choose x=2,…

Q: 5. Let n be a positive integer, and assume that a and b are integers such that a = b (mod n). Prove…

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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: 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: Recall that a = b( mod n) if a = b+ kn for some integer k. Use this to prove that if a = b ( mod n)…

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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: 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: Does 5x = 6 mod 12 have a solution? How many?

A: As per our guidelines we are supposed to answer only One question  if there are multiple questions…

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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Q: 0 4

A: 4 is true