1. Compute the following via the division algorithm (i.e. n = dq + r with 0

Q: 4965 4 mod 5 4965 = 2 mod 7 4965 = 1 mod 11 Use your answers above to find the canonical…

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Q: 1. In this problem we are working over the field Z7, the integers mod 7. All arithmetic is done mod…

A: In this problem, we are working over the field Z7, the integers mod 7. We use all arithmetic is done…

Q: The goal of this exercise is to practice finding the inverse modulo m of some (relatively prime)…

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Q: (a) (i) Without using the Euclidean algorithm, write down integers s, t such that 1 = 4s + 9t, and…

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

Q: 5. Prove that m(n) + nº(m) = 1 (mod mn) if (m, n) = 1. %3D

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Q: Suppose , p € Z such that p is prime. If (mod p) = 1 (mod p) find the least positive integer e such…

A: We have find least positive integer  such that .

Q: 1. Use the binary exponentiation algorithm (mod 1 537) 537)

A: Introduction: Binary exponentiation (also known as squaring exponentiation) is a technique that…

Q: Prove that for all integers x, if x is not divisible by 7, then x^3 ≡ 1 mod 7 or x^3 ≡ −1 mod 7.

A: Since you have asked multiple questions, we will solve the Question Q2 for you. If youwant any…

Q: Find 96307 7 (mod 739) using the techniques described in Example 8.4.4 and Example 8.4.5. First…

A: To find 96307mod739. It can be computed by evaluating the following,…

Q: 12. Use the Extended Euclidean Algorithm to find the multiplicative inverse of 23 mod 111.

A: Since you have posted a multiple question according to guildlines I will solve first(12)   question…

Q: In this whole worksheet, suppose n₁ = 3 and n₂ = 5. 1. Compute N = ₁+₂= Write out the list of…

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Q: Theorem 4.12. Gauss: Let n #0 (mod p). Consider the remainders mod p of the following (p-1)/2 many…

A: Let n≢0mod p. Consider the remainders mod p of the following p-12 many integers, n, 2n,......,p-12n.…

Q: For some of the most interesting laws of modular arithmetic (Choose some of your liking, but do…

A: This is a problem of mudular arithmatic.

Q: 2. If a = 3 (mod 11) and b = 9 (mod 11), compute the values, positive and fully-reduced modulo the…

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Q: 7. Calculate each of the following. [from #22, 4.5] b) 971321 (mod 765) a) 2557341 (mod 5681)

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Q: Provide the definition for each of the following terms, where n22 is an integer. (1) Congruent…

A: Given that n≥2 is an integer.

Q: Use 672 = -1 mod 449 and Hermites' algorithm to find a pair of positive integers s and t such that…

A: Given :          s2 +t2 = 449672 = -1 mod449

Q: 3. Show that ifn = 2 (mod 4), then 9" + 8" is divisible by 5.

A: Here we will prove the given statement.

Q: Let m be the number of these remainders which exceed p/2. Then ( #) = (-1)m.

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Q: 1.16. Do the following modular computations. In each case, fill in the box with an integer between 0…

A: We have to calculate the following modulo.Note: According to the guidelines we are allowed to solve…

Q: A. Show that if gcd(a, 65) = 1 then a 1 (mod 55). В. Use the binary exponentiation algorithm to…

A: A) Given that, gcda,65=1 To find : a24≡1mod 65 Fermat's theorem states that ," If P is a prime…

Q: (4) Apply the Fast Modular Exponentiation to find 0 < M < 87, where 5015 = M mod 87.

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Q: 5. Show that 2 is a primitive root mod 61 and compute log₂ (29) mod 61.

A: To show 2 is a primitive root we have to show that it has order 60. We do this and illustrate the…

Q: Compute (2100 mod 3), (2100 mod 5), and (2100 mod 7). Explain how you got the answers.

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Q: Use the EXTENDED EUCLIDEAN ALGORITHM to compute the following mul- tiplicative inverses: (a) 17-¹…

A: The extended Euclidean algorithm is an algorithm to compute the integers x and y such that for given…

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: 6. Show that ifn = pq, a product of distinct primes, then a (n)+1 (mod n) for all a. = a

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Q: 6. Trace Algorithm 4 when it is given m = 7, n = 10, and b = 2 as input. That is, show all the steps…

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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: Solve for d, 19d ≡ 1 mod (37422000) using the "Extended Euclid's Algorithm"

A: The solution is given by using Extended Euclid's Algorithm as follows

Q: Recall the Fibonacci numbers Fn are defined by Fo = 0 , F1 = 1 and Fn Fn-1 + Fn-2 for all n > 2 .…

A: Consider the provided question, Fibonacci numbers Fn are defined by F0=0, F1=1 and Fn= Fn-1+Fn-2 for…

Q: Define the following operation on Zn: { [1] if a = b (mod 5) [0] if a zb (mod 5) (a] * (b] = (a) Is…

A: Binary Operation : Let S be a non-empty set, then a map f : S×S →S defines a binary operation.…

Q: Find s e Z with 0ss< 13, so that s = 127+300(mod 13). Your work should show a method that does not…

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Q: The goal of this exercise is to practice finding the inverse modulo m of some (relatively prime)…

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Q: Find the smallest non negative value of D such that ED is congruent to 1 (mod b) when b=616…

A: As given in the hint the idea is to use the extended Euclidean algorthim to solve the congruence.…

Q: 11.5. For each part, find an x that solves the given simultaneous congruences. (a) x = 3 (mod 7) and…

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Q: Exercise 6.3.4. Find all solutions modulo 10 to x² + 3x = 0 (mod 10). There are two distinct…

A: We employ the chinese remainder theorem to find the four solutions. Details typeset in latex in step…

Q: 5- Let consider two (02) numbers a and b with the binary expansion a=10110011 and b= 101001 what is…

A: Since you have posted multiple questions and we can answer one question. So answered first question.

Q: 3. Use Euler's Theorem (and the Chinese Remainder Theorem) to how that n12 = 1 (mod 72) for all (n,…

A: The Euler's Theorem states that for a positive integer n and an integer a that is relatively prime…

Q: 2016 What is the highest power of 15 that divides | 1000 Be careful! 15 is NOT prime.

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

Q: Show that if n divides m where m and n are positive integers greater than 1, then a = b (mod m)…

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Q: 3. Find the last two digits of the number 99°. [Hint: 9° = 9 (mod 10); hence, 99 = = 99+10k. notice…

A: In question number 3: we have to find the last two digits of the number (((9)^9)^9) That means to…

Q: 8. Show that 18!= -1 (mod 437). 9. Using Euler's Theorem, find the last two digits of 44443333 10.…

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