1. Let p be an odd prime. Let a be an integer such that x2 = a (mod p) has that a (mod p has exactly two solutions for all a solution. Then prove positive integers k

Q: Prove that for all integers n ≥ 1, 10" = (-1)" (mod 11). Use part (a) to prove that a positive…

A: (a) Let n≥1 be an arbitrary integer. We know that 10≡10mod 11≡-1mod 11. Now consider 10n:…

Q: Find the smallest positive integer k such that 3k = 1 (mod 19).

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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: Show that if p is an odd prime, then 2(p - 3)!= -1 (mod p) Prove that if n has an odd number of…

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Q: a p−1 ≡ 1( mod a) for any integer a and prime p. Select one: True False

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Q: 1. Show that k1601 = k (mod 17) for any k.

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Q: Find all values of n with 0 ≤ n ≤ 35 such that the congruence 24x ≡ n (mod 36) has a solution.

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

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Q: Let x be an integer. Prove that, if mod3=2, then 3|(x2-1).

A: I hove used the theorem on congruent

Q: (a) Let n be any integer. Prove that n2 is congruent to 0, 1 or 4 modulo 8.

A: Note: According to bartleby guidelines we have to answer only first question please upload the…

Q: Supose you have a number 12340 that has a prime factorization of 22*5*617. What are the solutions to…

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Q: Let p be a prime with p ≡ 3 (mod 4), and suppose that q = 2q + 1 is also prime. Determine if 2 is a…

A: let p be a prime with p≡3mod 4 and suppose that q=2p+1 is also prime determine if 2 is a square mod…

Q: Solve the congruence 11x = 45 (mod 97) where x is an integer in the interval [0,97).

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Q: 4. Let p be a prime number and suppose that p = 3 (mod 4). Let A be any integer. Prove that exactly…

A: To prove that exactly one of the congruences X2≡A(modp) and X2≡−A(modp) has a solution, where p is a…

Q: - 1)/2 many integers: n, 2n, P=¹n. ∙n. 2

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Q: Let p be a prime number. The polynomial p(x) = x²+x+1 is irreducible over Zp if and and only if…

A: Given that for a prime number p, the polynomial px = x2+x+1 is irreducible over ℤp if and only if…

Q: Let p be an odd prime and a, b E Z such that (a, p) 1 and (b, p) = 1. Prove that if neither x² = a…

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Q: Prove that an odd integer n is a square modulo every power of 2 if and only if n ≡8 1.

A: Given n is an odd integer. We have to prove that an odd integer n is a square modulo every power of…

Q: We'll now try to prove the following result in 2 steps: Theorem: Suppose that p is an odd prime. Let…

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Q: Suppose that p≥ 5 is prime. Show that -3 is a quadratic residue (mo p) if p = 1 or 7 (mod 12) and a…

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Q: 7. Prove that if n is an odd positive integer or divisible by 4, then 13 +23 + 33 + .+ (n– 1)3 = 0…

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Q: Let p be prime and a an integer not divisible by p. Prove that if a²"-1 (mod p), then a has order…

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Q: 1.34. This exercise begins the study of squares and square roots modulo p. (a) Let p be an odd prime…

A: Given:(a) Let be an odd prime number and let be an integer with does not divide . (b) (i) (ii)…

Q: 3. Exercise 55.2 #12. Let p be an od«

A: Given, p be an odd prime number.

Q: Give nonlinear equation:

A: Given,             sinx=cotx and x0=1, and x1=0.5

Q: Let a be an integer. Prove the following statements: (a) If a is even, then a 2 ≡ 0 (mod 4). (b)…

A: We have to prove given results:

Q: DuppOse that a anа т аге intege Such that a, аre al diпerent IMod- ulo m, with am-1 = 1 mod m. Show…

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Q: Let p be an odd prime number. Prove that [(3)=1] → [p = 1 mod 6).

A: Introduction: An integer-based arithmetic system that takes the remainder into account is called…

Q: Find all values of m with 0 ≤ m ≤ 23 such that 10x ≡ m (mod 24) has a solution

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Q: (a pour) = (d(1) a) +...+ d& T dG T dl) +04+ mos ad ppo desta sedang

A: Here we use that (p-i)^p=-(i^p) in mod p. We proof this by binomial expansion.

Q: Prove that for any prime, p = 1 (mod 4) there are exactly (p-1)/4 incongruent values of a 0 (mod p)…

A: By theorem (a,p)=1 then xn≡a modp has g.c.d(n,p-1) solution or no solution according as…

Q: b) Determine whether or not x? = 85 (mod 97) has a solution.

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Q: Prove that for any integer x, if x mod 3 does not divide 0, then (x2 −3x + 1) mod 3 = 2.

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Q: 7. Let p and q be primes and suppose that p = 2q + 1. Let h E F and assume that h 1. Pre that ² h…

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Q: Prove: Let p be a prime number and m EN such that m <p². Then if p = 3 (mod 4) then the equation m =…

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Q: Suppose that p is a prime number and g is primitive root. Prove that the least residues of 4 9²,9ª,…

A: Step 1:To prove that the least residues of g2,g4,g6,gk,...,gn−1 are all quadratic residues modulo n,…

Q: Show that the Bernoulli number B₁ satisfies B₁ = (-1)" ( 2 + 2d²+1) Σ (mod 1) for any positive…

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Q: Determine all solutions x ∈ Z to the triple x ≡ 5 (mod 6), x ≡ 7 (mod 10) and x ≡ 2 (mod 15) of…

A: We reduce the congruence and then solve

Q: Suppose p is an odd prime. Show that (1² +2 +3P +...+ (p − 1)²) = 0 (mod p).

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Q: Ulia primeive Toute 95. Find all primes p ≤ 29 such that 2 is a primitive root modulo p.

A: Here we have to find all primes less than 29 or equal to 29 having primitive root 2.

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: e) Let p and q >7 be primes such that 2 +2P +1 =0 mod q. Prove that q = 1+6kp for ome natural number…

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Q: Determine the possibilities for the final digit of a sixth power of an integer.

A: The last digits of a number are 0,1,2,3,4,5,6,7,8 and 9. The number 0 raised to any number will be 0…