Which of the following is not valid congruence Modulo n? Select one: O a. 12 = (7mod - 5) O b. 12 = (7mod5) O c. 12 = (7mod – 10) O d. None of them

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: The multiplicative inverse of 19 (mod 141) * O is 51(mod 141) None of these O is 19(mod 141) is…

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Q: 6. Let m, n EN with m, n > 2, and let a, b E Z. Prove that a = b (mod mn) implies a = b (mod m) and…

A: (a) a = b (mod mn)       => there exists some integer k such that                    a = b +…

Q: The prime number 8191 is a None Inverse of 7 modulo 11 Carmichael umber Mersenne prime mmber

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Q: Which of the following is the correct congruence modulo form of this statement, "If December 8, 2023…

A: Follow the procedure given.

Q: Solve the congruence 2x + 3 = 11 (mod 7). (Select all congruence classes below (modulo 7) that…

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Q: Let a, be Z and n E N. For each statement decide whether it is necessarily true, or whether it can…

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Q: Show that 2 is a primitive root modulo 3" for all n > 1.

A: To prove that 2 is a primitive root modulo 3n for all n≥1.

Q: Let x be an integer coprime to 1450 that is not congruent to 1 modulo 1450. Suppose we know that x1…

A: The given two MCQs are exactly same. Hence we will just solve the first one. And we will use the…

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: 2. If a = 1 (mod p), then a=1 (mod p). Select one: True False

A: If , then . The truthfulness of the given statement can be proved by taking an arbitrary value for…

Q: Solve the following congruences. Make sure that the number you enter is ≥0 and ≤N−1, where N is the…

A: According to the given information, it is required to solve the following congruences. For part (i)

Q: If a =b (mod m) and a = b (mod n) then a = b (mod m+n) Select one: O True O False

A:

Q: Hello, I need help for the problem in the picture. Thank you!

A: We are given that α ∈ Z is a primitive root modulo an odd prime p. So,    αϕ(p) ≡ 1 mod p => αp-1…

Q: 1- Write the congruence classes of modulo m. 2- Write the addition table (Cayley’s Table) of m=7. 3-…

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

Q: The number p = 65537 = 210 +1 is known to be prime; you may take this as given. Let g be a primitive…

A: Since, g is a primitive root modulo p, order of g is (p-1) = 216 Now, we know order of gk is…

Q: If a = b (mod m) and a = b (mod n), and gcd (m,n) = 1, then a = b (mod mn) Select one: O True O…

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

A: p=61 incongruent modulo 61={x:1≤x≤n with gcd(x,n)=1}={1,2,3,4,5,6,7,8,9,10,11,12,13,...60}

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 prime number 8191 is a None Mersenne primте тmber Carmichael umber Inverse of 7 modulo 11

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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: a. Let m be an integer such that m mod 5 = 2, what is 8m mod 5? b. Let n be an integer such that n…

A: Consider the given information: Let m be an integer such that m mod5 = 2. Then m=5k+2   where…

Q: Al. b. Find the inverse of 17 modulo 23, if exists. If not, give reasons.

A: The objective is find the inverse of 17 modulo 23 if exists or else give the reason why it does not…

Q: SImalest posiuiy OI the n) is order of a modulo n. Find the additive orders of each of the following…

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Q: Determine the cycle or lifetime (N) of the following congruential generator. Xi+1 = (13Xi + 9) mod…

A: Generally linear Congruantial generator is given as,  Xi+1 = (aXi + c ) mod m  Then it has period or…

Q: (a) Suppose c is an integer. If c mod 7 = 5, what is 5c mod 7? (b) Write () in factorial form. (c)…

A: Since there are two questions posted, we will answer the first question (1). If you want the second…

Q: a) Define what it means for a and b to be congruent modulo 7. b) Which pairs of the integers…

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Q: from the definition of congruence modilo n, determine if the following congruenxe is true or not:…

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Q: 2. Create the following tables: (a) Addition table for modulo 7 (b) Multiplication table for modulo…

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Q: The ISBN of a book is a unique 13-digit code', also known as ISBN-13, that encodes information about…

A: ISBN: 978-0-75664-979-1 The congruence relation is…

Q: If a = 1 (mod n) and b = 1 (mod n) then a - b = 0 (mod n) اختر واحدا: 2 3

A: True.

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

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Q: 6. Which of the following is not a property of congruence modulo n? A. If a = e mod n and beZ then a…

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Q: Let p > 3 be a prime. Show that if p2+ p + 1 is also prime, then p =5 mod 6.

A: We prove the result by the method of contradiction.

Q: You and your friend Lara are considering the linear congruence ax = b (mod m), where m = 6 x 11 x…

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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: The number of incongruent primitive roots modulo a prime p is: a) Ø(p) b) Фф — 1) с) р—1 O a O b

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Q: In the following sets of simultaneous congruences, find x1 x = 2 (mod 5) x = 4 (mod 7) x = 5 (mod…

A: None of these.

Q: Which one of the following recursive definition is to find the Greatest Common Divisor? f(a,b) =. b…

A: This is a problem of Number Theory and Coding.

Q: Prove that if n is an odd integer, then n^2 is congruent to 1 modulo 8.

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Q: In the following sets of simultaneous congruences, find X2 x = 2 (mod 5) x = 4 (mod 7) x = 5 (mod…

A: Use the Chinese Remainder Theorem.

Q: If a is a primitive root modulo n, then a¯1 is a primitive root modulo n O False True

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Q: Which of the following congruence is TRUE? a. 2≡5 mod4 b. 76≡31 mod9 c. 1≡1 mod3 d. 18≡3 mod9

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Q: In the following sets of simultaneous congruences, find x1 x = 2 (mod 5) x = 4 (mod 7) x = 5 (mod…

A: d) 11

Q: Let a, n ∈ N. Show that if an − 1 is prime, then a = 2 or a − 1 is prime and n = 1 or n is prime.…

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Q: Exercise 14.4.13. Fill in the blanks with the appropriate variables in the following proof of the…

A: The blanks are filled in text highlighted by red color.

Q: 36. List all the elements of G* . G = Z/36Z

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Q: 1 Prove each of the following assertions: (a If a = b (mod n) and m |n, then a = b (mod m). (6) If a…

A: We know that  a ≡ b (mod n) is equivalent to n | (b-a)

Q: 19. Which of the following pairs of integers is NOT congruent to modulo 9? 1 and 10 4 and -32 -2 and…

A: Given  19) Congruent to modulo 9 20) 24 is congruent to 58 mod n 21) 1220(mod 25)