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

Q: 2p a) n! = 2 (mod p), where for 0 <k < n. %3D k!(n - k)!

A: To prove the given part (a) 2pp≡2mod p where nk=n!k!n-k! for 0≤k≤n

Q: 7) Show that there are no primitive roots modulo 21 using EGLF and CRT.

A: Given value of n is 21 . Now, break 21 into its factor, 21=7×3 ,7,3=1 If m&gt;2 and n&gt;2 then the…

Q: Let a, b be integers. What does it mean for a = b (mod 17)? Describe the elements of Z17. Compute…

A: a ≡ b (mod 17) means that a − b = 17k for some k ∈ ℤ. ∴  b − a = −17k = 17(−k); henceb ≡ a (mod…

Q: k) Prove that x = 5108 = 179 (mod 433) is a solution to the equation x² + 1 = 0 (mod 433), and use…

A: 52 ≡25(mod 433)54 ≡625≡192(mod 433)55 ≡960≡94(mod 433)56 ≡470≡37(mod 433)512 ≡1369≡70(mod 433)524…

Q: Show that 2 is a primitive root modulo 3" for all n > 1.

A: To prove that 2 is a primitive root modulo 3n for all n≥1.

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

A: Remainder of the Polynomial

Q: Determine the smallest integer k > 1 such that ak = 1 modulo 2465 for all in- tegers a which are…

A: A composite number m is called Carmichael number if for all integers b coprime with m, bm-1≡1modm…

Q: Show that x2 = Y mod p has

A:

Q: Let p be an odd prime number and an integer ga primitive root with respect to mod p. So is the…

A: We will use basic knowledge of elementary number theory to solve this question.

Q: For integer n>1, nZ={nx:x in z}, nZ is set of all integer multiples of n. Prove that integer p is…

A:

Q: Let p be a prime divisor of the Fermat number Fn= 22^n+ 1 i. Show that ordp2 = 2n+ 1 ii. Use…

A: Given p be a prime divisor of the number Fn=22n+1. (i) Since p divides Fn=22n+1, therefore: Fn ≅0…

Q: 1. Prove that if ā and be Zn are invertible modulo n, so is ā - b. Is this the case for ā + b?…

A:

Q: Show that: Ifp is odd prime of the form 4k+1, then find a solution of x +1 0 mod p

A: We shall use Euler’s Criterion here. Euler’s criterion states as follows: “Let p be a prime. The…

Q: x4 >is ........in Q[x] Not prime ideal ONone of the choices O Prime not maximal

A: Answer is maximal ideal , i.e , last option , and proof is given below.

Q: He have to Show that lon = (-1)" (mod 1") shall use mathematical moluction Let Pin) := lon a (47"…

A:

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

A:

Q: 1. Show that if m and n are odd integers, then m+n is an even intege

A:

Q: (i) Prove that if p is a prime, then (p - 1)! = -1 (mod p). (ii ) Show that 18! = -1 (mod 437).

A:

Q: 4. The least positive residue of 2^1000000 mod 17 is: * 1 None of these

A: Question 01: By Division algorithm,  10,00,000= 16*62,500 and since 17 is prime and 17 is not…

Q: 5. Prove that the equation ø(n) = ¢(n+ 2) is satisfied by n = 2(2p – 1) whenever p and 2p – 1 are…

A: Euler totient function formula: p is prime then ϕ(p)=p-1 ϕ(pk)=pk-pk-1     ∀k∈ℕ   If gcd(a,b)=1…

Q: Let be a rational prime, with p = 1(mod 4). Prove that p is the product of two Gaussian primes that…

A: We will use the basic knowledge of number theory to answer this question properly and completely.

Q: a) Show that 2340 ≡ 1 (mod 11) by Fermat’s little theorem and noting that 2340 = (210)34.b) Showthat…

A: According to the given information, Fermat’s little theorem says that:

Q: (a) Suppose that rị and r2 are quadratic non-residues modulo an odd prime p. Does x = r¡r2 (mod p)…

A: Quadratic Residues: Let m,p=1. m is said to be the quadratic residue mod p if the following equation…

Q: 1. Suppose m, nEz* are relatively prime. Prove that for all a, bEZ, a =b (mod mn) if a =b (mod m)…

A: Note that: a is congruent to b modulo m if and only if m divides (a-b).

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

A: We have solve given problem:

Q: suppose a,b ∈ ℤ. Prove that a ≡ b (mod 10) if and only if a ≡ b (mod 2) and a ≡ b (mod 5).

A: In Solution we use divisibility formula that  If a and b are integers, a divides b if there is an…

Q: show that 2 is a primitive root mod 3^e for all e>=1

A:

Q: Let n∈Z. Prove that n3−7n+18 is divisible by 6 for every interger n. Do not use induction.

A:

Q: Least residue of 45^37 mod 19 when using Fermats Little Theorem

A:

Q: Number theory

A: Consider the provided question,

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: Using Fermat's Little Theorem 62003 (mod 11) Compute the last two digits of…

A:

Q: 2. Find the largest negative x € Z that simultancously satisfies 4x = 9 (mod 5) 2x = 0 (mod 6) x =0…

A: Given : 4x = 9 (mod 5) 2x = 0 (mod 6) x = 0 (mod 4)

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: 2. Find an inverse of 2 modulo 17 that is positive and less than 17.

A:

Q: (b) Give an example of a composite integer n such that the congruence x =1 (mod n) pas more than two…

A:

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.

A:

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

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

Q: If n 2 1, then the Fermat number Fn 22" +1 is prime if and only if %3D 3(F.-1)/2 = -1 mod (Fn).

A:

Q: 10. Use Euler's Theorem to show that 5P(16), '=1 (mod 16). Show your work or explain your reasoning…

A:

Q: Use Wilson's Theorem to prove that for primes p congruent to 1 modulo 4 the quadratic congruence 1'…

A: Given The quadratic congruence x2≡-1(mod p) To prove  x2≡-1(mod p) has a solution if p≡1(mod 4)…

Q: Does x = a (mod p) always have a solution for every value of a, whenever p is prime?

A:

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: 3. a) Compute Ek. b) Prove that there are infinitely many prime numbers. c) Assuming modulus 127,…

A: Given: The following conditions. To find: a) Compute ∑k=0nk3 b) Prove that there are infinitely many…

Q: The least positive integer x satisfying 22010 = 3x (mod 5) is

A:

Q: Exercise 25.2. Prove that log, (bc) = log, b+ log, c (mod p- 1) for all b, c not divisible by p.

A:

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

A: We find solution of given congruence relation

Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.