Find s e Z with 0ss< 13, so that s = 127+300(mod 13). Your work should show a method that does not rely on finding the remainder of 427 modulo 13. Show your work.

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: Use the Chinese Remainder Theorem to find a solution to the congruences ´x = 2 mod 10 x = 3 mod 1001…

A: The given system is, x≡2mod10x≡3mod1001x≡3mod9 Chinese reminder theorem: If m1,m2,…,mn are positive…

Q: Find a polynomial p(x) with degree 6 and integer coefficients, such that for all integers x, we have…

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Q: Suppose b is any integer. If b mod 12 = 5, what is 6b mod 12? In other words, if division of b by 12…

A: Given b is an integer such that b mod12=5 To find the value of  6b mod12

Q: 8. Find the remainder of x2021 + x2020 + 1 modulo x2 – 1 in Q[x].

A: Remainder of the Polynomial

Q: As a small illustration of Chinese remainder theorem that can all be solved mentally, say M = 30.…

A: The Chinese Remainder Theorem:                               Let m1,m2,.......mk be a collection of…

Q: Determine the coefficient of at in (1 – 4a²)ó by completing the following given steps.

A: Concept:

Q: suppose a prime p of the form 8n + 3 and that q = (p-1) / 2 is also prime. Prove that ordp (2) = 8n…

A: Consider the prime p is of the form 8n+3 and q=(p-1)/2 is also prime. Then, objective is to prove…

Q: Modular division can be performed by considering the related multiplication problem. For instance,…

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Q: 2X=3 mod 5 Show your work

A: the given congruence equation is 2x≡3mod 5 the general congruence equation is ax≡bmod c…

Q: Problem 1: (a) Compute 13-1 (mod 23) by enumerating multiples. Show your work. (b) Compute 13-1 (mod…

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Q: If p &gt; 2 is a prime of the form 2m +1, prove that m is a power of 2.

A: Let p be a prime and p &gt; 2. And, p = 2m +1 Suppose 't' is an odd prime factor of m. That is, m =…

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: (b) Since 61 is a prime with 61 = 1 (mod 12) the polynomial X? + 3 has roots in Z/61Z by (a): they…

A: Here we need to solving ....(i) knowing that .....(ii)

Q: 5. Use the method of successive squaring to compute 3875 (mod 103) 1

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Q: b. Find the remainder using Fermat's Little theorem 452 mod 11.

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Q: Suppose that p≥ 5 is prime. Show that -3 is a quadratic residue (mo p) if p = 1 or 7 (mod 12) and a…

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Q: For p an odd prime in Z , Prove: If -1 is a perfect square in Z/pZ then p= 1(mod 4)

A: This can be done as follows:

Q: If you randomly pick integers from 1 to 200, how many would you have to pick to ensure that at least…

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Q: a. Find ∅ (171) b. Find the remainder using the Fermat’s Little theorem for 762 + 15 mod 13

A: As per our guidelines we are supposed to answer only one question (if there are multiple question…

Q: (b) By applying part (a), solve the congruences 2x = 1 (mod 31), 6x = 5 (mod 11) 3x = 17 (mod 29).…

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Q: 3. First solve for q: (a) 143 = 1( mod 2) (b) Rewrite 143 as the product of 11 x 13 and show that…

A: Consider the given information: a=q m+ r Where a= dividend , q=quotient, m=mod , and r=remainder. To…

Q: Find the inverse to 2 by using the Euclid's algorithm  2 mod

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Q: Find an integer A so that: A79 = 7 (mod 277) Please solve step by step thanks!

A: Given an equation. Let's solve for A.

Q: 2X+3=1 mod 5 Show your work

A: We have solve given problem:

Q: Euler's Theorem states that if n is a positive integer and a is relatively prime to n, then ar(n) =…

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Q: 1. Solve the congruence 17x ≡ 42 (mod 100). 2.Compute the remainder of 3293 when divided by 97…

A: 1) solve the congruence 17x=42+100k implies17≡42+100k as we know gcd17,100=1 and 1/42 now, using…

Q: Find all values of m with 0 ≤ m ≤ 23 such that 10x ≡ m (mod 24) has a solution

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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: Question 1a: Use Fermat’s Little Theorem to calculate: (i) 23106 (mod 107)  (ii) 431  (mod 7) (iii)…

A: As per our guidelines we supposed to solve the very first question if multiple questions are posted,…

Q: 4. a. Using the Euclidean algorithm, find the greatest common divisor, d, of 55 and 34 and solve the…

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Q: Show that for any odd prime p, the Gauss, G(p), can be rewritten as p-1 G(p) = 5². x=0

A: To Show- Show that for any odd prime p, the Gauss, Gp, can be rewritten as Gp = ∑x=0p-1ζpx2.

Q: The remainder when aa is divided by 77 is 22 and the remainder when bb is divided by 77 is 55. (So a…

A: Given:          a≡2 mod(7)  and b≡5 mod(7)

Q: Given the following system (M). (x = 2(mod 14) x = 1(mod 15) x = 3(mod 16) x = 2(mod 17) Which among…

A: Concept:

Q: If 2n + 1 is prime for some n ≥ 1, prove that n is a power of 2.

A: We have given 2n +1 is prime for some n ≥ 1 We have to prove that n is a power of 2.

Q: 2. Find all solutions, or show there are none, to the congruence X78 = 5 (mod 19)

A: By Fermat little theorem x18≡1mod19. Therefore, x4×18+6≡x6mod19 or x78≡x6mod19.

Q: Given the following system (M). (x = 2(mod 13) x = 1(mod 14) x = 3(mod 15) x = 2(mod 16) Which among…

A: According to the Chinese remainder theorem for a system of equation  x≡a1 mod n1x≡a2 mod n2⋮x≡ak mod…

Q: Let n = pq be the product of two odd primes. Recall that there are no primitive roots modulo n. Show…

A: We use the idea that if primitive root exists then x^2 = 1 mod (m) has only 2 solutions. The…

Q: Suppose b is any integer. If b mod 12 = 7, what is 5b mod 12? In other words, if division of b by 12…

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Q: (2) Let p be an odd prime and suppose that m e Z is not divisible by p. (1) Show that E() \p- = 0.…

A: (1) Given that p is an odd prime. The aim is to show that ∑a=1p-1ap=0. Now take, ∑a=1p-1ap,…

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

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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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Q: Q7 Compute 59^11 mod 13 Note: You should be able to do this using the modulo squaring method and…

A: 59^11 mod13

Q: 4) The least positive integer residue of 21000000 mod 17 is a) - 1 b) 1 c) 2 d) a

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Q: Suppose a # b (mod m). Let r be the remainder of a when divided by , and let s be the remainder of…

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Q: (b) Since 61 is a prime with 61 = 1 (mod 12) the polynomial X2 +3 has roots in Z/61Z by (a): they…

A: Given that, 61 ≡ 1 mod (12) The polynomial x2 + 3 has roots in Z/61Z. The roots are 27 + 61Z and -27…

Q: Find the solution to 2x = 7(mod 17) that is positive and less than 17.

A: We find solution of given congruence relation

Q: 6. Show that if n (a2" - 1)/(a² - 1), where a is an integer, a > 1, and p is an odd prime not…

A: The given question is related with pseudoprimes. We have to show that if n = a2p - 1a2 - 1, where a…

Q: Problem 6. Find all positive primes p such that n = 1001p satisfies that a" = a mod n for all a.

A: