If p > 8 is a prime number. a is an integer so that a4 ≡ 2 (mod p).  what is the least residue of a(p-1) + 12 mod p

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

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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: - Calculate the orders of 2 (mod 67) and of 2 (mod 73).

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Q: If mx = 1 (mod n) has a solution, then nx = 1 (mod m) has no a solution. Select one: True False

A: We know the definition of Congruence ,  Suppose m be a positive integer , we say that integers a and…

Q: The notation x = y mod n is used to indicate the unique integer x in the range 0 :$ x < n such…

A: Given statement is that the notation x = y mod n is used to indicate the unique integer x in the…

Q: 4. For any integer n, show that if n³ = 1 (mod 47), then n = 1 (mod 47).

A: To show that if n3≡1(mod47), then n≡1(mod47), it can use proof by contradiction.Assume that…

Q: 2. If a = 1 (mod p), then a=1 (mod p). Select one: True False

A: If , then . The truthfulness of the given statement can be proved by taking an arbitrary value for…

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: 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: If a =b (mod m) and a = b (mod n) then a = b (mod m+n) Select one: O True O False

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Q: 355 ( mod 7) + 234 ( mod 7)

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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: 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: 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: 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: 5. Let n be a positive integer, and assume that a and b are integers such that a = b (mod n). Prove…

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Q: (ii) The first few cubes are 1,8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331, and 1728. Write 1729…

A: The number 1729 is known as a Carmichael number. It can be represented as the sum of two cubes. For…

Q: Compute (2100 mod 3), (2100 mod 5), and (2100 mod 7). Explain how you got the answers.

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Q: If you randomly pick integers from 1 to 200, how many would you have to pick to ensure that at least…

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Q: (a) Suppose that rị and r2 are quadratic non-residues modulo an odd prime p. Does x = r¡r2 (mod p)…

A: Quadratic Residues: Let m,p=1. m is said to be the quadratic residue mod p if the following equation…

Q: Suppose that a ≡ 9(mod 13) and b ≡ 7(mod 13). Find the integer cwith0≤c≤12suchthatc≡2a2 −6b2(mod13).

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Q: Recall that a = b( mod n) if a = b+ kn for some integer k. Use this to prove that if a = b ( mod n)…

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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: 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: p is a prime number; a is an integer; Let (a, p) = 1 and the exponent of a with respect to mod p is…

A: please see the next step for solution

Q: Najabhai

A: The objective of the question is to find all integers n that satisfy the equation n² + 503 = 0 (mod…

Q: 10. Prove that a³ =a (mod 6) for every integer a.

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Q: The least residue of 5500 mod(17) is Select one: O none O 7 O 13 O10 O 8

A: The least residue is, 13 The detailed solution is as follows below :

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

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Q: b) Prove that if n2 (mod 3) = 1 then n is not divisible by 3.

A: The required proof is as follows -

Q: (d) For all integers a and b, if ab = 7 (mod 12), then either a = 1 (mod 12) or a = 7 (mod 12).

A: (d) For all integers a and b, if ab=7 mod 12, then either a =1 mod 12 or a =7 mod 12

Q: Find all solutions of x² = 27 (mod 253). Note that 253 is not a prime.

A: We have to find all solutions of x2 = 27 (mod 253).

Q: Please write neatly and provide all the necessary steps for solving If p is an odd prime prove that…

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Q: Let a be an integer. Prove the following statements: (a) If a is even, then a² = 0 (mod 4). (b) If a…

A: Congruence: If a and b are integers then a ≡b mod n for some n >0 if and only if n| b - a.

Q: True or False Let n > 0 be fixed and a, b, c and d are arbitrary integers, then, if a ≡ b(mod n)…

A: We have to check whether the statement, "Let n>0 be fixed and a, b, c and d are arbitrary…

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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