1. Let p be prime and let a 0 (mod p). Let b = a 2 (mod p). Show that ab = 1 (mod p). 2. Use the method of part (a) to solve 2x = 1 (mod 7).

Q: Which of the following equations has integer solutions? 1) 14x + 35y = 93 2) 33x + 14y = 115 3) 6x +…

A: Linear Diophantine equation ax+by+c=0 ∀a,b,c∈ℤ a,b are not both zero A solution of this equation is…

Q: Number Theory

A: Since we only answer up to 3 sub-parts, we’ll answer the first 3. Please resubmit the question and…

Q: If n is a positive integer with (n + 1)(n + 3) is odd, then (n + 2)(n+ 4) must be a multiple of 1) 6…

A: Given:  (n+1)(n+3) is odd We will get an odd number from product of two numbers only if the two…

Q: If a has order hk, then the order of ak is : а) hk b) h c) k O b O a

A:

Q: Number theory

A:

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: Number theory

A:

Q: Number theory

A:

Q: Number theory

A:

Q: Number theory

A:

Q: Number Theory

A: Given that n is a positive integer. The objective is to prove that

Q: If p is a prime and k is any integer, then S = {1*, 2*, 3*, .., (p - 1)*} is a reduced residue…

A: A subset S of integers is called a reduced residue system modulo p if : (i) gcd(a,p)=1 for each a∈S…

Q: Question The following linear Diophantine equation : 60x + 33y = 9 is solvable and its solution is:…

A:

Q: Number theory

A: Consider the provided question,

Q: Find the (general) solution of the following system by Chinese Remainder Theorem. 3x = 7 (mod 5) T =…

A:

Q: If a has order n-1 mod n, then n is prime. O True O False

A:

Q: The Gcd(5a + 2,7a + 3) is a) 2 b) a c) 1

A:

Q: The linear Diophantine equation 24x + 18y = 42 is solvable. False True This option This option

A:

Q: Number theory

A: According to the given information, it is required to find multiplicative inverse of 4 modulo…

Q: Number theory

A:

Q: Question * The cube of any integer may has the form: 9k + 8 9k + 7 This option This option 9k + 6…

A:

Q: Which of the following statements is true: For all integers a, b and e, if (a + b)lc and then a | c…

A:

Q: Number theory

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

Q: If a is a primitive root modulo n, then a¯1 is a primitive root modulo n O False True

A:

Q: Number theory

A: According to the given information, k is any number and n is a positive integer. Now denote the…

Q: The Legendre Symbol (-) is equal to 100 1 1 O None of the choices

A: Legendre symbol using properties

Q: Number theory

A:

Q: Number theory

A:

Q: Number theory

A:

Q: Number theory

A: Assume a=x2

Q: the following polynomial irreducible over Q? (Prove your statement)! 14x6 – 18x3 + 9x² + 6x – 60 - -

A: The polynomial is irreducible over Q by using Eisenstein's Criterion for irreducibility test.

Q: Number Theory

A: According to the given information it is required to compare the two number by using mediants.

Q: Find 21234567 (mod 509822653).

A: The objective is to evaluate 21234567mod 509822653. Obtain the prime factorization of 509822653. The…