Problem 1: (a) Compute 13-1 (mod 23) by enumerating multiples. Show your work. (b) Compute 13-1 (mod 23) using Fermat's Little Theorem. Show your work. (c) Compute 11-11 (mod 19) using Fermat's Little Theorem. Show your work. (d) Use Fermat's Little Theorem to compute 71209643 (mod 11). Show your work. (e) Find an integer x, 0≤x≤ 40, that satisfies 31x + 42 = 4 (mod 41). Show your work. You should not use brute force approach. (f) Calculate 138-1 (mod 2784) using any method of your choice. Show your work. (g) How many integers have inverses modulo 144? Justify.

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: Problem 9. (a) Compute 16! (mod 17). (b) Show that 12!= 12 (mod 221). You may use the fact that 221…

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Q: 6. Find the units' digit of 202220212020

A: We will use the following points to figure out the units digit in this question: Let's assume 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

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Q: Instructions: Comprehensively solve/explain the following. Show complete solutions/explanations. 1.…

A: We shall use wilsons theorem and euler's theorem to solve this problem. The detailed solution is…

Q: Problem 1. Give an alternative proof for Gauss;s Theorem by using that O is multiplicative.

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Q: (b) Use the fast factoring algorithm to compute 213 (mod 29).

A: (b) Exponentiation property: AB(mod c)=A mod cB(mod c) Since 213 is a really large number we split…

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: Compute 9-¹1 (mod 19) using Fermat's Little Theorem. Show your work.

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Q: For the purposes of fast modular exponentiation, compute 3183 mod 17 by instead computing the…

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Q: Find x in each of the cases below. I need a good explaination and understanding on how to use…

A: powermod:  Syntax: c=powermod(a,b,m) c=powermod(a,b,m) returns the modular exponentiation ab mod m.…

Q: (a) Write 23 as a sum of distinct powers of 2. (b) Use the fast factoring algorithm to compute 323…

A: To represent 23 as a sum of distinct powers of 2, we can use the binary representation of the…

Q: (c) Find integers p and q so that 17p + 45q = 1. Show your working steps. (d) Using the result in…

A: Given that 17p+45q=1 has no solution such that p and q are integers which means gcd of (17, 45) is 1…

Q: Y OU (a) Find the least residue of 639 modulo 10. (b) Use Fermat's little theorem to find the least…

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Q: If m = 2* is a power of 2, explain how you could use repeated squaring to compute am (mod n) for all…

A: a) We have m = 2k. First we illustrate the method of repeated squaring to compute am (mod n). Then…

Q: 5.PT. PLN has $140000 to invest. The alternative investments and the corresponding annual returns…

A: According tot he given information it is required to form an linear programming problem for the…

Q: Work the modular arithmetic problem. (11.10)(mod 7). (11 10) (mod 7) = (mod 7)

A: (a)   (11.10)mod 7  =  110 mod 7                            The remainder when 110 is…

Q: Question 2 Make sure that all your answers are explained carefully. (a) Find the remainder of 117…

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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: (a) Prove the following: i. 11 + (6 x 4") - 22n = 6 mod 7, 1 is divisible by 131, ii. 8260 iii.…

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Q: Find the smallest positive integer a that solves the congruence: 9x 6 (mod 58)

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Q: (b) Find the remainder of (10100) (10+00) mod 13.

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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: explanation

A: Round-Robin tournament: This comes out from the round-robin principle, where each person gets an…

Q: 2.3(i) Find the conjugates, moduli and arguments (to the nearest degree) of the following

A: 2.3(i)   (a). Given,                         3+5i.     To find the conjugate, modulus and argument…

Q: (a) Let p be an odd prime. Give an analogue of the quadratic formula mod p, and prove it. (b)…

A: (a) Given that p is an odd prime.  The quadratic formula mod p is given by: ax2+bx+c≡0mod p Here, p…

Q: Work the modular arithmetic problem. (8.9)(mod 4). (8.9)(mod 4) = (mod 4)

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Q: (a) Write 13 as a sum of distinct powers of 2. (b) Use the fast factoring algorithm to compute 213…

A: 13 can be written as a sum of distinct powers  of 2. 2^3=8 2^2=4 2^0=1  2^3 + 2^2 +2^0

Q: (c) Find all integers of order 6 mod 13 and all primitive roots mod 13. Show your working in both…

A: This is a question of Number Theory.

Q: 3. Find the residue of 20211202 mod2022 using Euler's Theorem.

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Q: (8) (a) Write 13 as a sum of distinct powers of 2. (b) Use the fast factoring algorithm to compute…

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Q: Work the modular arithmetic problem. (12 9)(mod 3). (12 9)(mod 3) =(mod 3)

A: The given modular arithmetic problem is -  (12·9)(mod 3) We have to solve the above problem.

Q: Problem 4 Use fast modular Show all steps used to arrive at your answer. exponentiation to compute…

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