Exercise 1: Modular arithmetic Explain how to find the smallest integer m>0 such that the congruence class of m coincides with the congruence class of... 1. ... 123456789 modulo 2.

Q: Find the number of elements in the conjugacy class of (abc)(def) in S₆

A: We have to find the number of elements in the conjugacy class of (a b c)(d e f) in S₆

Q: a) Give an example of a number m possessing exactly 12 primitive roots. b) How many primitive roots…

A: As per company guidelines I have solved first three sub parts if you want to get answer of the 4th…

Q: For each of the following primes, find the smallest positive integer k such that k is a prim root…

A: A primitive root of a prime p is an integer g such that g(modp) has multiplicative order p-1.

Q: (Discrete Math) Use Euclid’s algorithm to calculate gcd(33+35,33²+35²).

A: The GCD of a pair of a number can be calculated using Euclid's algorithm. The algorithm uses a…

Q: (a) Find three distinct integers satisfying the equation r = 1 mod 11. (b) Prove that if k is an…

A: a≡bmod m means a-b is divisible by m where a and b are integers. For example 13≡3 mod 5⇒13 - 3 is…

Q: Show that if n is odd and 3 n, then n2 = 1 (mod 24).

A: The solution is below.

Q: Prove that the order of 3 modulo 19 is a divisor of 18.

A:

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: Determine all integers n > 1 for which (n-1)! is a zero-divisor in Zn.

A: Given: Zn. To Do:We have to determine all integers n>1 for which n-1! is a zero divisor in Zn.

Q: Test2

A: We know that2198=2128×264×24×22Also we have22≡4(mod 397)24≡16(mod 397)28≡256(mod…

Q: (a) Prove that 2 and 3. are both primitive roots modulo 5.

A:

Q: Apply the Extended Euclidean Algorithm to find the multiplicative inverse of a mod n, provided it…

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: 1- Write the congruence classes of modulo m. 2- Write the addition table (Cayley’s Table) of m=7. 3-…

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

Q: 1. a) Determine whether binary operation is associative and whether it is commutative or not * is…

A: a) Given, a*b=a-b Now, a*b=a-bb*a=b-a So, a*b≠b*a So, * is not commutative Given, a*b=a-b Now,…

Q: Perform the following congruence computations. Make sure that the number you enter is ≥ 0 and ≤ N-1,…

A:

Q: *discrete math* Prove that for all integers a, b, if a and b are congruent to 1, mod 3, then their…

A: An integer n is congruent to 1, mod 3, if and only if there is an integer k such that n = 3k + 1.

Q: 2 has order m modulo (2m - 1).

A: To prove that 2 has order m modulo , we need to show two things:For any positive integer

Q: Perform the following congruence computations. Make sure that the number you enter is 20 and ≤N-1,…

A: (a) 5946+7925≡        mod 45⇒13871≡        mod 454530813871-13500371-360011∴5946+7925≡  11      mod…

Q: If gcd(7, a) = 1, prove that either a +1 or a – 1 is divisible by 7. 20

A:

Q: A binary string is a word containing only Os and 1s. In a binary string, a 1-run is a non-extendable…

A: def count_1_runs(n): binary_representation = bin(n)[2:] # get the binary representation of n…

Q: 24. Show that 1P + 2P + 3P + .+ (p - 1)P =0 (mod p) when p is an odd prime. Show that if nis prime…

A: Using Wilson's Theorem, Fermat's Little Theorem and The Pollard Factorization Method, we have to…

Q: (a) Use cases based on congruence modulo 3 and properties of congruence to prove that for each…

A:

Q: (b) Prove that if a = 7 mod 10 and b = 6 mod 15, then a + b = 3 mod 5.

A: Our objective is to prove . Since you have posted multiple questions I am solving one.

Q: 28. Show that if p and q are distinct primes, then p-1+qP-=1 (mod pq).

A: To prove: If pand q are distinct primes then, pq-1+qp-1≡1(mod pq).

Q: Let A [1 0 7 ]5=[-5 3 21* and ne following, or show that it does not exist. (o) dot( RC)

A:

Q: List all of the reduced (i.e. invertible) residue classes modulo 24.

A: To find all the reduced (invertible) residue classes modulo 24, we have to find integers n such that…

Q: Prove: Any positive integer can be represented as an aggregate of different powers of 3, the terms…

A:

Q: 13 [The induction step can skip through the integers] Prove these divisibility results: a n2 + 2n is…

A:

Q: Do the divisors of 24 form a Boolean algebra under gcd and lcm?

A:

Q: 3) Let the order on the X=(-∞0,5√2) NQ be the known order. find the supremum Supremum and infimum of…

A:

Q: a² Prove that if ged(a, m); 1 and gcd (a-1, m)=1, then 1+a+a² + ... + a(m) = 0 (mod m). (Hint: What…

A: (a−1)(1+a+a2+⋯+aϕ(m)−1)=aϕ(m)−1≡0(modm)Now…

Q: Number theory

A: Determine the number N in terms of u,t,h,s,….