a) Define what it means for a and b to be congruent modulo 7. b) Which pairs of the integers −11,−8,−7,−1, 0, 3, and 17 are congruent modulo 7? c) Show that if a and b are congruent modulo 7, then 10a + 13 and −4b + 20 are also congruent modulo 7.

Q: Prove or disprove the following: A) If a is positive integer and 7|a-3 , then 7|a²-9. B) Suppose a…

A:

Q: 1): For which values of n is n div 5 = -6? A): n = -30, -29, -28, -27, and -26. B): n = -30, -15,…

A: see below the answer and explanation

Q: 3) The possible values of the positive integer m such that 27 = 5(mod m) are: a) 2 b) 11 c) 22 d)…

A: We have to find m such that 27 = 5(mod m)

Q: Q// Let m = 5, 1- Write the Addition table (Cayley's Table). 2- Write the congruence classes of…

A: For m=5 , our set is Z5 .

Q: Prove that the order of 3 modulo 19 is a divisor of 18.

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Q: 2. Find the integer a such that a = 2*(mod31) and -16 < a< 15. а) 16 b) -16 с) -15 d) 15 e) 8

A: Here we have to find a.

Q: 5. Let a, b, c, and n be integers. Prove that if a = b (mod n) and c = d (mod n), then a − c = b -d…

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Q: Let m be fixed positive integer. Then an integers ais congruent to an integer b modulom, if: a-b|m C…

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Q: USE PENCIL and PAPER, DONT USE AI State as TRUE or FALSE. Show your work. A) 172 ≡ 2 (mod 5) B) -6 ≡…

A: Modular Arithmetic involves numbers wrapping around after reaching a certain value called the…

Q: a) Prove that every integer can be written a sum of powers of 2. b) Write 14, 17, 27 as a sum of…

A: We have to prove the every integer can be written a sum of power of 2 and write 14, 17, 27 as a sum…

Q: Determine whether 24 and 14 are congruent modulo 6. DONT USE AI, USE paper and pencil

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Q: Which expression is equivalent to [77 · (-65) + 147] mod 7? O 7 O [(147 mod 7) · (-65 mod 7) + (77…

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Q: a) If 4x1 = 4 mod 9 and 0 < x < 9, then x = b) If 70x + 19 = 0 mod 141 and 0 ≤ x < 141, then x = c)…

A: a) If 4x-1≡4 mod 9 and 0≤x&lt;9, then x= . b) If 70x+19≡0 mod 141 and 0≤x&lt;141, then x= . c) If…

Q: *discrete math* Prove that for all integers a, b, if a and b are congruent to 1, mod 3, then their…

A: An integer n is congruent to 1, mod 3, if and only if there is an integer k such that n = 3k + 1.

Q: Find an inverse modulo 13 of each of the following integers. a) 2 b) 3 c) 5 d) 11

A: if a number n has x (integer) as its inverse modulo 13 then n·x ≡ 1(mod 13) ⇒ n·x = 13k+1 here k is…

Q: 7) Use a direct proof to show that: a) The sum of an even integer and an odd integer is odd. b) If n…

A: Use the following concepts, to obtain the required result. An even integer m can be written as a…

Q: a) If a| (21b−c) and 6c≅3b(mod2a), then a| 41c b) If a≅b(modc) and d|c then ?≅?(mod?) c) ∴(d|…

A: this is a question of modular arithmatic.

Q: 4) If a is an odd integer, then a? = ?? (mod 8) a) 0 b) 1 c) 2 d) 3

A: We have to find a such that a2 = ??(mod 8)

Q: True or False: 5 is a primitive root modulo 29. O True O False

A:

Q: 2

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Q: Explain why all prime numbers greater than 3 are congruent to either 1 or 5, modulo 6. If you are…

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Q: 1) Prove that the numbers -197, -239 are prime numbers 2) Find all prime numbers between -90 and 90.

A: number system

Q: let a,b,c,d, and n be integers with n>1. Prove that if a is congruent to b(mod n) and c is congruent…

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Q: Test6

A: solution of the given question is like as:we use the legendry symbol and its property and quadratic…

Q: solve it math A, B,and C

A: Answer 1a)OR {x ∈ Z+|x≡1 (mod 8)}Answer 1b)Answer 1c)

Q: Suppose n is a natural number greater than 100. Alex and Jerry each compute the gcd(4n, 20n2 + 15)…

A: Given that n is a natural number greater than 100. Alex and Jerry computed the gcd4n,20n2+15 and got…

Q: - Let a, b, n be positive integers. Prove that if a = b (mod n), then (a, n) = (b, n).

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Q: Najabhai

A: The objective of the question is to find all integers n that satisfy the equation n² + 503 = 0 (mod…

Q: Prove that ifa≡b(modn)andb≡c(modn), thena≡c(modn). Note that you must use thedivides definition of…

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Q: 11) The solution of the linear congruences x = 1 (mod 2), x = 2 (mod 3), x = 3 (mod 5), x = 4 (mod…

A: We have to find value of x

Q: a) What does it mean for a to be an inverse of a modulo m? b) How can you find an inverse of a…

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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: proof that if a equiv b(mod n), then a+c equiv b+c (mod n) and ac equiv bc (mod n).

A: Apply the definition of congruence and divisibility, to express a≡bmod n. From the definition of…

Q: ie/False: The numbers 252 and –108 are congruent modulo 15. True False

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Q: Which of the following is valid congruence Modulo n? Select one: O a. None of them ОБ. 12 (-5тod3) O…

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Q: We say (a, b, c) is a Pythagorean Triple if a, b, c are positive integers and a? 3² = c². For…

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Q: b) Prove that if n2 (mod 3) = 1 then n is not divisible by 3.

A: The required proof is as follows -

Q: Jsing only definition of divisibility, definition of modulo, substitution, and arithmetic, prove for…

A: a ≡ b (mod m), c ≡ d (mod m) ⇒ m | a – b and m | c – d ⇒ m | ((a – b) + (c – d)) ⇒ m | ((a + c) – (b…

Q: 236,153 2 = | - Is 36,153 Yes

A: 36153To check whether it is composite or not.

Q: Show that : If n is an integer then n is congruent to 0 modulo 4 or 1 modulo 4.

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Q: True or False Let n > 0 be fixed and a, b, c and d are arbitrary integers, then, if a ≡ b(mod n)…

A: We have to check whether the statement, "Let n&gt;0 be fixed and a, b, c and d are arbitrary…