Suppose that p is a prime with p = 5 mod 8 . Show that the congruence px2 + 6y = 1 mod 3p has no solutions for integers x and y.

Q: What is the number of solutions to the congruence in Z125? x³ + x² + 3 = 0 (mod 125)

A: As per the question we have to solve the following polynomial congruence equation : x3 + x2 + 3 ≡ 0…

Q: Let p be a prime. Show that -2 is a quadratic residue modulo p if and only if p = 1 mod 8 or p = 3…

A: Definition(Quadratic Residue Modulo): In number theory an integer qis called a quadratic residue…

Q: The linear congruence in two variables ax + by ≡ c (mod m), where a, b, c, and m are integers, m > 0…

A: As we know-the congruence ax+by≡c (mod m)has dm incongruent solution if   d=(a,b,c)|c& no…

Q: 6. Let p be an odd prime. Show that a2 = a (mod p*), k > 2 has exactly two solutions if x2 = a (mod…

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Q: For congruences of primes p = 1 (mod 4), a. If a is a quadratic residue modulo p, and p = 1 (mod 4),…

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Q: For p an odd prime in Z , Prove: If p=(a)^2 + (b)^2  is possible in Z then   p= 1 (mod 4)

A: Compute general prime that satisfies given condition as follows.

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: Give an odd prime p congruent to ±1 modulo 12. Show that (3) = 1.

A: Please see the attachment

Q: If p is a prime with p=3 mod 4 and a is quadratic residue mod p, then -a is a quadratic residue mod…

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Q: Find all solutions in {0, 1,2, ..., 12} to x26 + 15x13 = 12 mod 13.

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Q: For which primes p > 5 is -5 a square mod p? Your answer should be of the formp = {a1, a2, ..., ak}…

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Q: For which odd primes p is 5 a square modulo p?

A: As we know that  a is Quadratic residue mod p if and only if   a(p-1) /2 =1 mod p . If p and q are…

Q: (9) Find all prime numbers p for which both equations X²-2 0 mod p and have solutions. X²-3 0 mod p

A: Concept: The quadratic congruence: x2≡a(mod p) , where gcd(a,p)=1 Has a solution if (ap)=1 That is a…

Q: (5) Show that if n ɛ Z+ is odd which is divisible by k distinct primes, then the congruence x2 = 1…

A: Chinese Remainder Theorem: If a1,a2,...,ak are integers, and m1,m2,...,mk are pairwise relatively…

Q: Exercise 4.21. Prove that x2 = 1 (mod p) has exactly two solutions when p> 3 is a prime. 70

A: We shall solve it using basic knowledge from elementary number theory, in particular the properties…

Q: The linear congruence 7x = 3(mod16) is solvable of solution: O None of the choices

A: The given problem is to solve the linear congreunce equation, we have to use congruence properties…

Q: Let p be a prime and a a quadratic residue of p. Show that if p =1 (mod 4), ais also a quadratic…

A: Let P be an odd prime and Let a be a quadratic residue mod p ∵ P-a≡-a (mod p) ∴(P-a)p-12≡(-a)p-12…

Q: 5. Let p be an odd prime, and suppose that ged(a, p) = 1. Show that the congruence z = a (mod p*)…

A: p is an odd prime and gcd(a,p)=1We have to show the congruence x2=a mod pk has exactly 1+ap…

Q: Prove that the number of solutions to the congruence xk ≡ 1 mod p is gcd(k, p − 1)

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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: Find all solutions mod 31 to the congruence z = -5 (mod 31).

A: We will use python code to find the solutions Now x5 ≡  -5 (mod 31)  ≡  26  (mod 31) And we only…

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

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Q: It is given that 3023 is prime, and 3023 = 2(1511) +1 where 1511 is also prime. (a) How many…

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Q: Suppose x satisfies the three simultaneous congruences X = 4 (mod 5) x = 7 (mod 8) X 10 (mod 11)…

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Q: 9. Show that if p is an odd prime and p = 3 mod 4 then p is not the sum of the squares of two…

A: In the given question we have to prove that if p is an odd prime and…

Q: Suppose that p and q = 4p+ 25 are both prime. Show that p is a quadratic residue mod q.

A: Given p and q are both prime. and q=4p+25 To show that p is a quadratic residue mod q.

Q: Find the first few terms of the sequence of pseudorandom numbers generated using the linear…

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Q: 2. Prove that for any m = 1 mod 11 that there are no integer solutions of m = 2r" - 9y.

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Q: Find the first few terms of the sequence of pseudorandom numbers generated using the linear…

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Q: I know that using a corollary that the congruence x2 ≡ 0 (mod p) that the only solutions are + and -…

A: As per the question we have the following congruence relation : n ≡ 2(mod 4) And we have to prove…

Q: 6), then the congruence has a unique solution modulo p. 11. Show that the congruence x³ = 3 (mod 19)…

A: 11. To prove that the congruence x3≡3 mod 19 has no solutions, whereas x2≡11(mod 19) has three…

Q: You and your friend Lara are considering the linear congruence ax = b (mod m), where m = 6 x 11 x…

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Q: Suppose p is a prime congruent to 3 modulo 4. (a) Prove that the congruence r = -1 (mod p) has no…

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Q: Find the first few terms of the sequence of pseudorandom numbers generated using the linear…

A: We have to find few terms of the given sequences.

Q: If mx = 1 (mod n) has a solution, then nx= 1 (mod m) has a solution. Select one: O True False

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Q: Find all solutions of the congruence x2 ≡ 16 (mod 105).

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Q: Use the index system modulo 19 to find all solutions of x³ = 11 (mod 19) and 5ª = 4 (mod 19)

A: In the theory of congruence, Fermat's little theorem is very significant. It says that for any prime…

Q: Show that if p is an odd prime and a is an integer not divisible by p, then the congruence x2 ≡ a…

A: According to the given information, Let p is an odd prime and a is an integer not divisible by p…

Q: Exercise 36 Prove that x? = a (mod p) has either no solutions or 2 solutions when p is a prime and a…

A: I have use the method of contradiction

Q: 5 is a quadratic non- residue mod 17

A: The quadratic residues mod17 are { 1,2,4,8,9,13,15,16}

Q: Q3/ Use Chinese theorem to solve the following congruence: X=2 mod 3 X=1 mod 5 X 2 mod 4

A: Since you have posted a multiple question ,I will solve the first question(i.e.Q3)  for you. To get…

Q: Find a two digit integer c such that 101c = 1 mod 107. This congruence has a unique solution mod 101…

A: We need to find a digit integer c such that 101c ≡ 1 mod 107. As per the given question, this…

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: (b) Prove that if p is a prime greater than 2 then 2(p- 3)! = -1 mod p %3D

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Q: Find all solutions to the congruence 10x≡5 mod 15 (b) Prove that 7 divides 32n+1+ 2n+2for any…

A: In  (a) we use the extended Euclidean algorithm to solve the problem. In (b) we use the mathematical…

Q: Number theory

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Q: 3. Find all integer solutions to the pair of congruences (if any) x = 1 (mod 2) x = 2 (mod3) x = 3…

A: To Find: Integer solutions to the pair of congruences(if any) x≡1(mod 2), x≡2(mod 3), x≡3(mod 5)…

Q: Determine a general congruence condition satisfied by all primes p with p = 3(mod4) such that the…

A: As per the question we are given a number theory problem. Where we have to find a congruence…

Q: Find the smallest positive integer xx that solves the congruence: 11x≡5(mod57)

A:

Statements and Logic

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