Using Fermat's Little Theorem, solve the congruence 2. x425 + 4 · x108 – 3 . x² + x – 4 = 0 mod 107. Write your answer as a set of congruence classes modulo 107, such as {1,2,3}.

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: (a) (i) Without using the Euclidean algorithm, write down integers s, t such that 1 = 4s + 9t, and…

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

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: k) Prove that x = 5108 = 179 (mod 433) is a solution to the equation x² + 1 = 0 (mod 433), and use…

A: 52 ≡25(mod 433)54 ≡625≡192(mod 433)55 ≡960≡94(mod 433)56 ≡470≡37(mod 433)512 ≡1369≡70(mod 433)524…

Q: Solve the congruence 2x + 3 = 11 (mod 7). (Select all congruence classes below (modulo 7) that…

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Q: For congruences of primes p = 1 (mod 4), a. If a is a quadratic residue modulo p, and p = 1 (mod 4),…

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Q: 6. (i) Solve the congruence relation 3x + 5 = 2(3 – 5x) (mod 35) for r. - (ii) If a = b (mod n) and…

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Q: 2. Let a be a positive integer. Explain why the linear congruence ax = b (mod a + 1) has a solution…

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Q: How do you fiNd all positive integers m for which the following congruence is true. 100 ≡ 50(mod m)

A: We know if a≡b( mod m) then a-b = mk where k is integer. Given,      100 ≡ 50 ( mod m ) We have to…

Q: Describe congruence conditions on p > 2 an odd prime that classify exactly when 11 is a quadratic…

A: We have to describe congruence condition on p>2 an odd prime that classify exactly when 11 is a…

Q: 1. Solve the following linear congruence using a systematic method: 167x = 241 (mod 199) Present…

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Q: The Modular Operation r mod m = r denotes that r is the remainder = 3. If two integers have the same…

A: Rewrite the following theorems using quantifiers and predicates.

Q: [Hint: Proceed by induction on n.] 16. Use the theory of congruences to verify that 89| 24 – 1 and…

A: The solution is given as follows :

Q: Suppose g is a primitive root mod p (with p an odd prime) such that g +1 mod p? Prove g must be a…

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Q: a) How can an inverse of a modulo m be used to solve the congruence ax ≡ b (mod m) when gcd(a,m) =…

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Q: Use Fermat's theorem, show that if gcd(a, 145) = 1 then a141 = a (mod 145).

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Q: Find the least positive integer n that satisfies the congruence 7 ^128 congruent to n (mod13)

A: Given, 7128≡nmod 13 Find the value of n that satisfies the above congruence.

Q: Use Fermat's Little Theorem to find (Show work.) a) 945 mod.23. b) 231002 mod. 41 c) (-9)933 mod. 13

A: Fermat's Little theorem states that " if p is a prime number and a is not divisible by p then…

Q: (b) By applying part (a), solve the congruences 2x = 1 (mod 31), 6x = 5 (mod 11) 3x = 17 (mod 29).…

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Q: brvib - If m is composite, what is the value (m – 1)! mod m in the standard residue system? Conclude…

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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: 1. (i) Show that 7 is a primitive root modulo 41. (ii) Find all solutions mod 41 of the congruence…

A: (i) to show 7 is primitive root modulo 41 proof a number b is said to be primitive root modulo P…

Q: (a) Find an r E Z that makes the following congruence true: 5x = 1 (mod 24). (b) Prove that there is…

A: Solution a: Given  5x≡1 mod 24 For x=1 5≡5 (mod 24) for x=2 10≡10 (mod 24) for x=3 15≡15 (mod 24)…

Q: 6. If c is a positive integer and a and b are integers with a = b(mod c), show that a2 = b²(mod c).…

A: Let c is positive integers and  a and b are integers with a ≡ b ( mod c)

Q: (a) Use cases based on congruence modulo 3 and properties of congruence to prove that for each…

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Q: 8. Let â be the inverse of a modulo k. Prove that the order of a modulo k is equal to the order of â…

A: If a is a primitive root modulo k , then for all n (coprime with k) there exists m , such that : am…

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: Suppose that (n, e) is an RSA encryption key, with n = pq, where p and q are large primes and gcd(e,…

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Q: 1. Use Euler's Theorem to prove Q265 = a for all a E Z. a (mod 105)

A: note : As per our company guidelines we are supposed to answer ?️only the first question. Kindly…

Q: Use Fermat's theorem, show that if ged(a, 155) 1 then a1 = a (mod 15

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Q: (a) Suppose p is prime, pła and k = gcd(n, p – 1). If p-1 a k =1 (mod p), prove that the congruence…

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Q: Suppose p is a prime congruent to 3 modulo 4. (a) Prove that the congruence r = -1 (mod p) has no…

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Q: (c) Find all integers of order 6 mod 13 and all primitive roots mod 13. Show your working in both…

A: This is a question of Number Theory.

Q: 4n, then 9| (unt1 ++1 e, establish the following: (a) 3 = Un (mod 2), hence u3, U6, Ug, ... are all…

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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

A: Since you have posted a multiple question ,I will solve the first question(i.e.Q3)  for you. To get…

Q: Q7 Compute 59^11 mod 13 Note: You should be able to do this using the modulo squaring method and…

A: 59^11 mod13

Q: Prove Exercise 4.6.12 in the course notes. Then, assuming that gcd(a, n)|b, determine how many…

A: We have to prove that if gcd(a,n) | b, then ax≡b (modn) has no solution. We will prove this by the…

Q: Use Wilson's Theorem to prove that for primes p congruent to 1 modulo 4 the quadratic congruence 1'…

A: Given The quadratic congruence x2≡-1(mod p) To prove  x2≡-1(mod p) has a solution if p≡1(mod 4)…

Q: Determine whether the following relations are true or false. Explain your results (a) 111 = -5 mod…

A: Dear Bartleby student,according to our guidelines we can answer only three subparts, or first…

Q: (b) Prove that if p is a prime greater than 2 then 2(p- 3)! = -1 mod p %3D

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Q: Let p be prime. Suppose ab≡0 mod p and a ≢ 0 mod p. Apply Euclid’s Lemma to prove b ≡0 mod p

A: Given that, Let p be prime. Suppose, ab≡0 mod p and a≢0 mod p. To prove: b≡0 mod p…

Q: Solve the congruence 200x = 13 (mod 1001). Give your answer as the smallest positive integer in its…

A: a = 200, m = 1001 Applying the Euclidean algorithm we get the inverse of 200 modulo 1001 is 996 (you…

Q: 2. (a) If gcd(a , 35) = 1, show that a2 = 1 (mod 35). [Hint: From Fermat's theorem a = 1 (mod 7) and…

A: Fermat's Little Theorem:- It states that if p is any prime and a is any integer then, ap≡amodp…

Q: Determine which of the following congruence relations are true and which are false. (a) 47 = 3 (mod…

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Q: 3. Find all integer solutions to the pair of congruences (if any) x = 1 (mod 2) x = 2 (mod3) x = 3…

A: To Find: Integer solutions to the pair of congruences(if any) x≡1(mod 2), x≡2(mod 3), x≡3(mod 5)…

Q: Find all three incongruent solutions to the congruence x³ = 1 (mod 103). Express these solutions as…

A: The solutions are obtained by knowing the definition of primitive roots and applying them carefully.…

Q: Suppose ab≡0 mod n and gcd(a, n) = 1. Apply Euclid’s Lemma to prove b≡0 mod n.

A: Given: ab≡0mod n and gcd(a,n)=1