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: Show that if a and b are integers and m and n are positive integers such that n|m, then If a = b(mod…

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Q: 8. Let m 2q + 1 be an odd positive integer. Prove that every integer = is congruent modulo m to a…

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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: Prove that 01 + 02 is even.

A: Any odd integer, p can be represented as p=2k+1 . Any even integer, q can be represented as q=2k .…

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

A: I hove used the theorem on congruent

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

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Q: a) Let p be an odd prime. Show that a = -1 mod p has a solution if and only if p = 1 mod 8.

A: Consider the provided question,Hello. Since your question has multiple parts, we will solve first…

Q: 3. Prove: for every integer n and for every odd integer a² = 1 (mod 2n+2)

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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: Prove that if x² = y² (mod n) and x #±y (mod n) then gcd(x + y, n) is a nontrivial factor of n.

A: We have,x2≡y2mod nAnd,x≢±ymod n. We have to show that gcdx+y, n is a nontrivial factor of n.

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: 11. Prove that a¹ = 1 (mod 5) for every integer a that is not divisible by 5.

A: We have to prove  for every integer  that is not divisible by

Q: Suppose that p is a prime with p = 5 mod 8 . Show that the congruence px2 + 6y = 1 mod 3p has no…

A: Consider the given information.

Q: Let (a,b)=1 with a = c (mod b) and bª = c (mod a) for some integer c. Prove that ab+(b). +ba+(a) = c…

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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: Let a be an odd integer. If a 3 we have (2ª) a 2 = 1 (mod 2°),

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Q: Let a and b be positive integers with (a, b) = 1. Show that if g is a primitive root (mod ab), then…

A: We have given (a,b)=1 where a, b both are positive integer. We have to show if g is primitive root…

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: m and n are relatively prime if and only if am+bn=1 for some integers a and b.

A: It is given that :am+bn=1--(1) for a,b,m,n ∈ Z Now, if  k divides both m and n, then there exist p…

Q: 16. Prove that an integer (a,-1an-2 · . . ao)10 is divisible by 11 if and only if ao + a2 + a4 + . .…

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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: 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: Let a € Z be odd. (a) Prove that a = 1 (mod 4) or a = 3 (mod 4). (b) Prove that a² = 1 (mod 4).

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Q: Let 1 ≤k≤p-1 where p is a prime number. Then prove that (₁7¹) = (-1) (modp) k

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Q: Let a and b be integers and n a positive integer. Assume also that a and n have a common divisor d…

A: Use the following concepts, to prove the required result. If a divides b then b is a multiple of a.…

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

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Q: 2. If the following statement is true, give a proof showing it to be true. If it is false, disprove…

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Q: Find an integer x satisfying the two congruences: x = 1(mod 3) 2x = 0(mod 6), or show that no such…

A: The given congruences are                                 x≡1 (mod 3)2x≡0 (mod 6)

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

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Q: a) Let a be an integer with a > 0. The integer a is even if and only if a = 0 mod 2. Note to self:…

A: (a) Let a be an integer with a > 0.       We need to prove that a is even if and only if a ≡ 0…

Q: 1. Let a and b be natural numbers, where a, b > 2, such that gcd(a, b) = 1 and ab = (x + y)" for…

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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: a is any integer and p is prime, then prove that a\power{p}≡a (mod p

A: If a is any integer and p is prime, then we have to show that

Q: 4. Let p> 2 be prime. Show that there exists an integer n such that n² + 1 is a multiple of p if and…

A: We use the fact that if gcd(a,p)=1 then a^{p-1}= 1 (mod p)