Generalization. The clock above is a mod 5 clock. Find ways to predict the output of mod 10, mod 2, mod 9, and mod 3 clocks.

Q: 3400 m is prime numbers, then compute 27 (mod m) 2. If m is natural numbers, show that a (mod m ) =…

A: Modular arithmetic: a≡b mod c means c divides a-b i.e., c|(a-b). So there exists a constant k such…

Q: 1) Find an integer a and an integer y, such that x = 5 mod 10 y = 7 mod 12 x = 9 mod 21 y = 2 mod 17…

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Q: Use Euler's Theorem to find 3112 (mod 23) None of the choices 3112 = 2(mod 23) This option O This…

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Q: Which one of the following can be used to find text message in RSA decryption mΞcd (mod n) m=c° (mod…

A: concept:

Q: Find the value of 592 (mod 11) using Fermat's Little Theorem. Remember, x (P-1) = 1 (mod p) a 592 =…

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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: If a = b (mod n) and a = c (mod n) then which of the following is true. (a) b = c (mod n) (b) b2 =…

A: We use the properties of congruences to answer the given question.

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

A: 1) Given expression is  49≡______ mod 10

Q: Question 6. Given integers a, b, c, d € Z, and m, n € N. (a) Suppose that a = b (mod n), and that m…

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Q: The n-th random number is generated by the function below. If xO=2, x2 is * Xn = (7Xn-1 + 9) mod 11…

A: Given Xn = ( 7 Xn-1 + 9) mod 11 X0 = 2 X2 = ?

Q: р-1 Suppose p = 1 mod 4 and g is a primitive root mod p. Show a = gʻ4 satisfies a? = -1 mod p.

A: p = 1 mod 4. g is primitive root modulo p. => order (g) = p-1.  => gp-1 = 1 mod p and gn ≠ 1…

Q: If a = b (mod n) and a = c (mod n) then which of the following is true. (a) b3 c (тod n) (b) b2 =…

A: Given below the detailed solution

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

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Q: (a) Give an inductive proof that the Fibonacci numbers Fm and Fm+1 are reatively prime for all m 20.…

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Q: ) Find all possible values of m: 53 = -7 mod (m)

A: m is positive greater than 1.

Q: Find the value of 786 (mod 13) using Fermat's Little Theorem. Remember, x (P - 1) = 1 (mod p) 786 =…

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Q: 5. Bob's RSA public key has modulus N = 391 and exponent 3. Alice sends Bob c = 213. You're Eve.…

A: The RSA algorithm is as follows: Key generation N=pq, where p and q are primes. select Exponent e,…

Q: 14. The expression "a mod b" is used to mean "the remainder obtained when a is divided by b". Which…

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

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Q: 4.19. Let a, b e Z. Show that if a = 5 (mod 6) and b = 3 (mod 4), then 4a + 6b = 6 (mod 8).

A: The detailed solution is as follows below:

Q: If m>4 is a composite integer, then (m-1)!=1 mod m (m-1)!=2 mod m (m-1)!=0 mod m None of these

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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: 1. Let n be a natural number with n > 1. (a) Suppose n is prime. Prove that if ab = 0 (mod n), then…

A: In 1)  (a) we need to prove that ab≡0 ( mod n), then a≡0 (mod n ) or b≡0 (mod n ).     In (b) We…

Q: 11. Here is our second code. Once more you agree with your friend on a modulus (say, 25) and a…

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Q: Q1 Correct 30.8s Question (334+458) mod 12 (Express your answer as an integer) Christina Agustin's…

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Q: B. Prove the assertions below: (a) If a is an odd integer, then a? = 1 (mod 8). (b) For any integer…

A: We will prove the assertions one by one. For the first part we will use the fact any odd integer a…

Q: If (a, 42) =1, then a^6-2=?mod 168 1 -1 None of these

A: Just apply the Fermat's Theorem.

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: 28. Show that if p and q are distinct primes, then p-1+qP-=1 (mod pq).

A: To prove: If pand q are distinct primes then, pq-1+qp-1≡1(mod pq).

Q: 5. Suppose you have a language with only the 3 letters a, b, c, and they occur with frequencies .7,…

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Q: III. Determine the value of a such that: 10. a = 11 (mod 21) and 90 s a s 110 11. a = 19 (mod 29)…

A: The given questions are related to congruences. The solution is explained below.

Q: VI. Determine four values of x that will satisfy the following. 16. x = 4 mod 5 & x = 3 mod 7

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Q: 3. The Lucas Lehmer Test is applied below to test whether the number 255 is prime or not. (S,' -2 k…

A: The Lucas Lehmer test: Let Mp =2p-1  be the Mersenne number to test where p is an odd prime.…

Q: III. Determine the value of a such that: 7. a = 20 mod 27 and -26 sa s0 8. a = 20 (mod 31) and –15 s…

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Q: If a has order 2k, then ak =? mod p a) 0 b) 1 c) -1 O b a

A: Solution

Q: O 1: If n is an odd, then solve (x- 1)(x +2) = (x - 2)(x+ 7) (mod n).

A: We’ll answer the first question since the exact one wasn’t specified. Please submit a new question…

Q: 30. Find the integer a such that a) a = 43 (mod 23) and -22 < a s0. b) a = 17 (mod 29) and -14 < a s…

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Q: III. Determine the value of a such that: 7. a = 20 mod 27 and –26 <a<0 8. a = 20 (mod 31) and –15 <…

A: The solution are next step

Q: (f) If a = b (mod n), then a* bk (mod n) for any positive integer k.

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Q: 12.55. Prove the following: Let a, b, m, n e Z, where m, n 2 2. If a = b (mod m) and a = b (mod n),…

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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: 20. If a e Z and a = 1 (mod 5), then a? = 1 (mod 5).

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Q: III. Determine the value of a such that: 7. a = 20 mod 27 and -26saso 8. a = 20 (mod 31) and -15 < a…

A: Please check step 2 for the solution.!

Q: 6) If p is an odd prime, then 1P + 2P + 3P + .. + (p – 1)P = ??(mod p) a) 0 b) 1 c) - 1 d) None of…

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Q: 6. Find a prime number p for which the equation x² = -1 mod p has no solution.

A: Given is a  modular equation.   To Find: A prime number p such that the given equation has no…

Q: 5) If p is an odd prime, then 2(p – 3)! = ?? (mod p) a) 2 b) 1 c) - 1 d) - 2

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Q: Question 6 Given integers a, b, c, d = Z, and m, n € N. (a) Suppose that a = b (mod n), and that m |…

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