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…

A:

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)…

A:

Q: b) 568 mod 7

A:

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

A:

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: 4444 is divided by a 1.19) find the remainder when 4444 * 2?=-1 (mod a).

A: 1.19 We have 4444  ≡  7 (mod 9). So, 44444444 ≡ 7 4444 (mod 9) Now,       73 ≡1 (mod 9) Or,…

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: 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…

A:

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: 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.

A:

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: - 1)/2 many integers: n, 2n, P=¹n. ∙n. 2

A:

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

A:

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.

A:

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: (d) Find an integer x, 0 < < 18, that satisfies the following congruence: 9x + 13 = 10 (mod 19).…

A: (d) Given is a congruence 9x+13≡10(mod 19)   To Find: An integer x such that 0≤x≤18 that satisfies…

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

A:

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

A:

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: 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…

A:

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

A:

Q: Sketch the graph of f by hand and use your sketch to find the absolute and local maximum and minimum…

A:

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)…

A:

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: Using appropriate methods, reduce (i) 419 mod 47, (ii) 141-10156 mod 47, (iii) 483 mod 47, (iv) (419…

A: We have to reduce the following congruent equation.

Q: 1 Using prime factorization, find gcd(900, 140) and lcm(900, 140).

A: Note: Hi! Thank you for the question as per the honor code, we’ll answer the first question since…

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

A: