15. Let r be a primitive root of n. If gcd(a, n) = 1, then the smallest positive integer k such that a = rh (mod n) is called the index of a relative to r, denoted by ind,a. The theory of indices can be used to solve congruences. Consider the properties of indices (p. 164) and example 8.4 (p. 164). Solve pI43? 8x = 11 (mod 13)

Q: 1.2. Problem 2. Let 1 < a < b be two positive integers. Prove or disprove: there exists a…

A:

Q: 10. Prove that if o is the m – cycle (a,a2 ... am), then for all i e {1,2, ...m}, o'(ar) = ar+i,…

A: Given that σ is the m- cycle (a1,a2,...am). We need to prove (i) σi(ak)=ak+i mod m for all i ∈{1, 2,…

Q: let x  be a fixed but arbitrary integer. Do there exist infinity many integer y such that,…

A: Let x = x0 be fixed given integer. Then,

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

A: Remainder of the Polynomial

Q: a) Prove that 10" =+1 (mod 11) for any n EN. b) Suppose the integer r has digits rI-1I, I0. Prove…

A: Want to prove:(a) 10n≡±1 (mod 11) for any n∈N(b) If the integer x has digits xnxn-1.....x1x0 thenx≡0…

Q: (1) Consider the equation 2.x +1 = 0 mod n. For which n are there no solutions? one solution?…

A: Congruence

Q: Let n and a be positive integers and let d = gcd(a, n). Show that the equation ax mod n = 1 has a…

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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: (4) Find the values of r in the following equations (such that 0 ≤r <n where n is the integer to the…

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Q: The number p = 65537 = 210 +1 is known to be prime; you may take this as given. Let g be a primitive…

A: Since, g is a primitive root modulo p, order of g is (p-1) = 216 Now, we know order of gk is…

Q: 2. Find x such that 43177 = x (mod 39), 0≤x≤39. You must show all the detailed steps.

A: see my solutions

Q: |State whether the following statements are true or false. Show working. a. For each odd prime p, a…

A: We have to state whether the statements are true or false

Q: . (a) How many quadratic residues are in the residue set Z19? (b) Determine if 4 and 9 are quadratic…

A: Introduction: Quadratic residue is a part of congruent modulo theory. It is a non-linear congruence.…

Q: and 10 9. Use Euler's theorem to evaluate 2100000 (mod 77). An11

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Q: Consider the pseudorandom number generator given in our lessons: with Uf +1 = (auf + b) mod m…

A:

Q: For every number a e a e {1, 2, 3, ..., 18} which is not a primitive root (modulo 19) find the…

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Q: 11- P, X Pz X P, ... X Pn, where M= Pr. Pn are distind primes Prove by inducton, For all Ppositive…

A: Fermat's little theorem is one of the most important and significant methods in number theory.…

Q: Suppose that phi (x) belongs to Aut (Zn) and a is relatively prime to n.  If phi (a)=b, determine a…

A: According to the given information it is given that:

Q: Use the identity 1/(k(k + 1)) = 1/k – 1/(k + 1) and Ex- ercise 35 to compute E- 1/(k(k + 1)).

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Q: (2) (a) Find all of the solutions modulo 15 to 6x = 15 mod 18. Express your answer in terms of the…

A: Given, 6x≡15 mod 18. As it is known that, an≡bn mod cn ⇔ a≡b mod c. Therefore, we have 6x≡15 mod 18…

Q: 1. Use PMI to show that for any n E N, 4 divides 5" - 1.

A: As per our guidelines, we are supposed to solve only first question. Kindly repost other question as…

Q: 7. Find a number t such that In(4t +3) = 5 (Give EXACT value)

A: Given that ln(4t+3)=5 we need to find "t"

Q: Problem 2. Let p be an odd prime number and let r be a primitive root (mod p). Thus, płr, rP-1 = 1…

A: Let p be an odd prime and r be a primitive root modulo p. ... Can Write A = R^k For Some K.  Prove…

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

A:

Q: Consider the following theorem. (") and is given by the formula Theorem 9.5.1: The number of subsets…

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Q: 6. Let Fn 22" + 1, n > 1, be a prime Fermat number. Show that 3 is a primitive root modulo Fn.

A: Given: Fn=22n+1,n&gt;1

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: 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: Is the following statement true or false? For every odd integer n, To answer this question, let n be…

A: Let, n be any odd integer. By definition of odd, n=2k+12, for some integer k So, by solution,…

Q: 1. Use Euler's Theorem to prove a 265 = a (mod 105) for all a E Z.

A: To prove:                   a265≡amod 105 for all a∈ℤ by using Euler's theorem.   Euler's theorem:…

Q: Show that th equation x" + y² = z" (mod 5) has nonzero solutions in Zz if and only if n is odd.

A: Given: xn+y2n≡zn(mod 5)

Q: Consider the subring S = 2216 of Z16. Then char (S) = 4 6. 8. оОО О

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Q: 1. Use Euler's Theorem to prove Q265 = a for all a E Z. a (mod 105)

A: note : As per our company guidelines we are supposed to answer ?️only the first question. Kindly…

Q: 12. Let the "Fibonacci-2" numbers g, be defined as follows: 91 = 2, 92 = 2, and gn+2 =( In+1)( In)…

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

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Q: Disprove the claim: 3 m E N such that m 2 3 and m? – 1 is a prime number. -

A: Given that ∃m∈ℕ such that m≥3 and m2-1 is a prime number. Take m=4. It is known that 4∈N And 4&gt;3.

Q: 4n, then 9| (unt1 ++1 e, establish the following: (a) 3 = Un (mod 2), hence u3, U6, Ug, ... are all…

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Q: Suppose n E Z. Prove that if n is even, then n² + 7n + 10 is even. (Use Direct Proof)

A: Suppose that n∈Z and let n is even.To show: n2+7n+10 is also even.

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

A:

Q: If g is a primitive root modulo 23, then for which 1 si< 23 is g' also a ive root?

A: Answer:

Q: 15. Prove the following identity C) + (" ; ') + (" * ) + (" ; ) +-+ " : )--- " * ** ') (n + k + k…

A: Binomial Coefficient. The binomial coefficient is the number of ways of picking unordered outcomes…

Q: Please find a constant M such that ,for any prime p,computing 1/a mod p(for any a exists…

A: Consider the provided question, We have to find constant M such that, for any prime p, computing 1/a…

Q: QUESTION 3 Prove that 2n > n2 if n is an integer greater than 4. For the toolhar press AIT. E10 /RCL

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Q: b) Prove that the following code returns the nth odd natural number by structural induc- tion. For…

A: To prove the given function generates n th odd primary number  These functions are called recursive…

Q: 2. (a) If gcd(a , 35) = 1, show that a2 = 1 (mod 35). [Hint: From Fermat's theorem a = 1 (mod 7) and…

A: Fermat's Little Theorem:- It states that if p is any prime and a is any integer then, ap≡amodp…

Q: . Let b, k and m be positive integers that satisfy gcd(b, m) = 1 and gcd(k, ø(m)) = 1, %3D where…

A: Here GCD(b,m)=1 means they are coprime.

Q: Consider the relations E9 and E12 on the set of integers defined as follows • aE9b precisely when…

A: Given relations aE9b precisely when a≡bmod 9 aE12b precisely when a≡bmod 12 where relation R=E9oE12.

Q: 4P- Problem 7. Show that for each prime p 2 5, the integer m, = " is a pseudoprime to base 2.…

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Q: In parts (a)- (e), involve the theorems of Fermat, Euler, Wilson, and the Euler Phi-function. (a)…

A: