MAT 255 Let p and q be distinet prime mumbers and let a be an integer not divisible by p and not divisible by q. Show that the congruence =a (mod p) has a unique solution modulo pg whenever the integer k is relatively prime to (p-1)(q-1).

Q: a) Prove that if p is prime and p  1≡ (mod 4), then ((p-1)/2)!)^2 ≡ -1 (mod p) b) Use part a) to…

A: According to the given information in part (a) it is required to prove that:

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 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: ​Q3// Use Chinese theorem to solve the following congruence: X ≡ 2 mod 3 X ≡ 1 mod 5 X ≡ 2 mod 4

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Q: Suppose , p € Z such that p is prime. If (mod p) = 1 (mod p) find the least positive integer e such…

A: We have find least positive integer  such that .

Q: Using Euclid's algorithm, find the multiplicative inverse of 23 modulo 75 and then solve the linear…

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Q: Q// Let m = 5, 1- Write the Addition table (Cayley's Table). 2- Write the congruence classes of…

A: For m=5 , our set is Z5 .

Q: Show that if p is an odd prime and a and b are integers with a ≡ b (mod p), then

A: According to the given information, let p be an odd prime and a and b are integers with a = b…

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

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Q: Congruence conditions (mod 20) for primes p for which -5 is a square (mod p).

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Q: Perform the following congruence computations. Make sure that the number you enter is ≥ 0 and ≤ N-1,…

A: Since you have asked a question with multiple  subparts so as per guidelines  we will  solve the…

Q: MAT 255 I. Let p and q be distinet prime mumbers and let a be an integer not divisible by p and not…

A: Note: Since you have asked multiple questions, we will solve the first question for you. If you want…

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

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Q: Perform the following congruence computations. Make sure that the number you enter is ≥ 0 and ≤ N-1,…

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Q: Number theory

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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: Congruence conditions (mod 20) for primes p for which -5 is a square (mod p).

A: The proof is based on a simple application of the law of quadratic reciprocity and the…

Q: Let a, n be positive integers. Knowing that 15a – 13n = 1, we deduce that the multiplicative inverse…

A: We have to find the moduler multiplicative inverse of a.

Q: Perform the following congruence computations. Make sure that the number you enter is 20 and ≤N-1,…

A: (a) 5946+7925≡        mod 45⇒13871≡        mod 454530813871-13500371-360011∴5946+7925≡  11      mod…

Q: True or False: 5 is a primitive root modulo 29. O True O False

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Q: 2) i) Show that 5 is a primitive root modulo 17. Find the all integers, less than 17, whose orders…

A: To show that 5 is a primitive root modulo 17, we need to demonstrate that it has the property of…

Q: Perform the following congruence computations. Make sure that the number you enter is ≥ 0 and ≤ N-1,…

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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: Find all natural numbers b with 100 < b < 200 such that: 5211 b (mod 35)

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Q: Consider the congruence 9x   21 (mod 33). (a) Does 9 -1(mod 33) exist? Explain why or why not.

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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: Give an example to show that a? = b? (mod n) need not imply that a = b mod n). fa=b(mod n) prove…

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Q: Prove for each integer n, that n2 is congruent either to 0 or to 1 modulo 4.

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Q: he smallest positive integer x such that x = 22 (mod 25), x =

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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: 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: 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: Determine the remainder of 102020 modulo 2020 and the remainder of 102020 modulo 2021.

A: solution 104≡1920(mod2020)…

Q: Apply the properties of congruence to make computations in modulo n feasible. Find the unique r such…

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Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.