Prove that the order of 3 modulo 19 is a divisor of 18.

Q: Find the number of elements in the conjugacy class of (abc)(def) in S₆

A: We have to find the number of elements in the conjugacy class of (a b c)(d e f) in S₆

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: Show that 2 is a primitive root modulo 3" for all n > 1.

A: To prove that 2 is a primitive root modulo 3n for all n≥1.

Q: 1) Compute the order of a modulo 7 for a = 1,2,....6. Find all primitive roots modulo 7.

A: Order of the element

Q: Prove that 2 is not a primitive root modulo 31.

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Q: Give a definition of an inverse of a number a modulo m.

A: if the two integer a and m modulo then  let x is an integer value so the x is inverse of an only if…

Q: Find the order of all element of Z8 under addition mod . Please brieflt explain and show all the…

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Q: Test2

A: We know that2198=2128×264×24×22Also we have22≡4(mod 397)24≡16(mod 397)28≡256(mod…

Q: Find the least residue of 448 modulo 7

A: Given problem is : Find the least residue of 448 modulo 7

Q: Find all integers between 1 and 20 that are zero divisors of mod 21

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Q: Find all the primitive roots modulo 125.

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Q: (2) Compute the order of a modulo 11 for a = 1,2,..., 10. Find all primitive roots modulo 11.

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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: Use a Proof by Contradiction to prove that there is no integer a such that a congruent 5 (mod 14)…

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Q: 8) Prove that 2 is a primitive root modulo 53.

A: We have to Prove that 2 is a primitive root modulo 53.

Q: Show that if a ≡ b (mod m) and c ≡ d (mod m), then a + c ≡ b + d (mod m).

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Q: Show that there are no integers a, b such that a² + 2ab+26² = 100000000003. Hint. Use mod 4…

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Q: 51 numbers are chosen randomly from the 100 integers 1-100, without repeating any number twice.…

A: The proof is as shown below.

Q: Prove that if x2 is congruent to y2 (mod n) and x is not congruent +or - y (mod n)  then gcd(x + y,…

A: Given: x2≡y2modn x≢±ymodn

Q: Find a number a such that 70a = 1 mod 17.

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Q: Compute (1001 mod d) for d = 2, 3,..., 10.

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Q: Show that y = x (mod p) leads to -y = -x (mod p).

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Q: Prove: Any positive integer can be represented as an aggregate of different powers of 3, the terms…

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Q: Give a recursive definition for the set of all the positive integers that are less than 100 and not…

A: Given: Set of all positive integers that are less than 100. To find: Recursive definition for the…