4. Let p be an odd prime, and a, b primitive roots mod p. Show that ab is not a primitive root mod p.

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: (()) b) Use part a) to find an integer that satisfies the congruence x² = -1 (mod 17) a) Prove that…

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Q: V. Suppose that a and b are integers, a = 7 (mod 19), and b = 5 (mod 19). Find the integer c with 0…

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

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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: a)Consider the following data: 3 is a quadratic residue for 23 and 37. 3 is a quadratic nonresidue…

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Q: Determine a positive integer x such that 57x +7 = 78 (mod53).

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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: Is 3 a primitive root of 43? Use your answer to determine all positive integers less than 43 whose…

A: To determine whether 3 is a primitive root of 43, we need to check if it generates all the positive…

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: x² + 4x = 10(mod11) (mod11) X=

A: Given that x2+4x≡10 (mod 11)x2+4x+4≡10+4 (mod 11)x2+4x+4≡14 (mod 11)x2+4x)(x+2)2≡14+11 (mod…

Q: Use Hensel Lifting (twice) to find a solution to x² = 7 (mod 374).

A: According to the given information, it is required to use Hensel lifting method to find the…

Q: Find all integers sx such that 2x = 5 (mod 7).

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Q: Let a, b, and c be integers and n a natural number. If a ≡ b mod n then a+c ≡ b+c mod n. How do I…

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Q: Using Wilson's theorem, to show that if p is a prime and p = 3 (mod 4), then ()! = +1 (mod p).

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Q: Prove that for any integer a, at = 0 (mod 5) or aª = 1 (mod 5)

A: To prove for any integer a,  a4≡0mod5 or a4≡1mod5.

Q: If p is a prime, then (p-2)!≡1(mod p). Select one: True False

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Q: 4. Find 2001 (mod 205). (Enter the least non-negative integer

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