20.1 Theorem (Little Theorem of Fermat) If a = Z and p is a prime not dividing a, then p divides aP-11, that is, aP-1 = 1 (mod p) for a 0 (mod p). 20.2 Corollary If a Є Z, then a² = a (mod p) for any prime p. Find the remainder of 5130 modulo 28

Q: 3400 m is prime numbers, then compute 27 (mod m) 2. If m is natural numbers, show that a (mod m ) =…

A: Modular arithmetic: a≡b mod c means c divides a-b i.e., c|(a-b). So there exists a constant k such…

Q: 6. Let m, n EN with m, n > 2, and let a, b E Z. Prove that a = b (mod mn) implies a = b (mod m) and…

A: (a) a = b (mod mn)       => there exists some integer k such that                    a = b +…

Q: 7.26. Theorem. Let p be a prime and a be an integer. Then a² is not a primitive root modulo p.

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Q: Suppose p is a prime such that p ≡ 1 (mod 4). Show that the sum of the quadratic residues mod p…

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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: .Which of the following integers are the possible one’s digits of a fourth power of any integer? a.…

A: # as per the guidelines we are entitled to solve one question at a time, please resubmit the other…

Q: 2. Give an example to show that a? = b2 (mod n) need not imply that a =b (mod n). 3. If

A: Given below the detailed solution

Q: Let be P odd an prime s.t- gcd (y² + 4v₁ p) = 1. 2 If (x + y √B ) ² = u ² + 4v (mod pk) (VK 6 N),…

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Q: 4.75. Let n, m E Z. Prove that if n =1 (mod 2) and m = 3 (mod 4), then n² + m = 0 (mod 4).

A: The detailed solution is as follows below:

Q: Prove that for all integers x, if x is not divisible by 7, then x^3 ≡ 1 mod 7 or x^3 ≡ −1 mod 7.

A: Since you have asked multiple questions, we will solve the Question Q2 for you. If youwant any…

Q: Give an odd prime p congruent to ±1 modulo 12. Show that (3) = 1.

A: Please see the attachment

Q: 1. Prove (2p+1) = 2 (mod p) when p is any prime.

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Q: Theorem 4.12. Gauss: Let n #0 (mod p). Consider the remainders mod p of the following (p-1)/2 many…

A: Let n≢0mod p. Consider the remainders mod p of the following p-12 many integers, n, 2n,......,p-12n.…

Q: 1.Let a, b, and c be integers, where a≠0. If a | b and a | c, then a | (b + c)            True or…

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

Q: Suppose m, n e Z† are relatively prime. Prove that for all a, b e Z, a = b (mod mn) iff a = b (mod…

A: The congruence relation a≡b mod n implies that a-b is a multiple of n. That is n| a-b or a=kn+b If…

Q: 7. Calculate each of the following. [from #22, 4.5] b) 971321 (mod 765) a) 2557341 (mod 5681)

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Q: 7. Show that if n is odd, then p(4n) - 2 q(n). =

A: 7. Show that if n  is odd,then phi(4n)=2phi(n)

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: Find the four solutions of x* = 1 (mod 13).

A: Since you have asked multiple questions we will solve the first question for you .To get any…

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

A: It is given that, p be an odd prime, and a and b primitive roots mod p.

Q: Let m be the number of these remainders which exceed p/2. Then ( #) = (-1)m.

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Q: 4. Let p be a prime number and suppose that p = 3 (mod 4). Let A be any integer. Prove that exactly…

A: To prove that exactly one of the congruences X2≡A(modp) and X2≡−A(modp) has a solution, where p is a…

Q: 5. Let p be an odd prime, and suppose that ged(a, p) = 1. Show that the congruence z = a (mod p*)…

A: p is an odd prime and gcd(a,p)=1We have to show the congruence x2=a mod pk has exactly 1+ap…

Q: 3. Use Wilson's Theorem to prove 6(k - 4)! = 1 (mod k:), if k is prime.

A: Wilson's Theorem: Let k be a prime number, then by Wilson's theorem, (k-1)!=-1(modk)

Q: (b) Wilson's Theorem: Let p be prime. Then (p-1)!= -1(mod p). Hint: Verify for p = 2 and p = 3…

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Q: Suppose that p≥ 5 is prime. Show that -3 is a quadratic residue (mo p) if p = 1 or 7 (mod 12) and a…

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Q: Suppose x F and you discover that x¹ = 1 mod p and x² = 1 mod p. Please prove that xgcd(m,n) = 1 mod…

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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: 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: Let p and q be two distinct primes. (a) Prove that if a = b (mod p) and a = b (mod q), then a = b…

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Q: 1. Let p be an odd prime and g be a primitive root mod p. Show that Z = {gik : k = 1,...,p-1} if and…

A: please find the attached image.

Q: a) Let p be prime. Suppose a, b e Z and ab=0 (mod p). Prove that either a=0 (mod p) or b=0 (mod p).

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Q: 7.27. Theorem. Let p be a prime and let i and j be natural numbers with i ‡ j satisfying 1<i,j<.…

A: Suppose, i2≡j2 (mod p)⇒i2-j2 ≡0 (mod p)⇒(i+j)(i-j)≡0 (mod p)⇒p|i+j or p|i-j⇒p≤i+j or p≤i-j

Q: 5. Prove that the integer 53103 + 10353 is divisible by 39, and that 111333 ible by 7.

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Q: 1. Let p be prime and let a 0 (mod p). Let b = a 2 (mod p). Show that ab = 1 (mod p). 2. Use the…

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Q: 4. Use Fermat's factorization method to factor 2168495737.

A: To use Fermat's factorization method to factor the below number. 2168495737

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: Problem 2. Let p be an odd prime number and let r be a primitive root (mod p). Thus, płr, rP-1 = 1…

A: Let p be an odd prime and r be a primitive root modulo p. ... Can Write A = R^k For Some K.  Prove…

Q: 1. Let p be an odd prime. Let a be an integer such that x2 = a (mod p) has that a (mod p has exactly…

A: 1. Let p be an odd prime. Let a be an integer such that x2 = a (mod p) has that a (mod p has exactly…

Q: Show that if a is a quadratic residue (mod p) and ab = 1 (mod p) then b is a quadratic residue (mod…

A: Note that quadratic residues means those which are squares mod p using this definition we solve the…

Q: 5 (9 9 () 2. Find a reduced residue system modulo 2", where m is a positive integer. in a reduced…

A: To find a reduced residue system modulo 2m, where m is a positive integer.

Q: 3 23) Give a counter example to disprove the foilowing Property ab = ac (modn) →6=C (modn) yb=C…

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Q: 0001E JO du (u pou) 0=1-(4)"+ ... 4. Show that if a and m are positive integers with (a, m) = (a –…

A: Given a and m are positive integers with a,m=a-1,m=1

Q: - Show that (a + b)" = a + bn mod 2 for all integers a, b, and n with n > 0. Hint: Try looking at…

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Q: 38. It is known that 14 is a primitive root for the prime p = 30000001. Let 14⁹000000 (mod p). (The…

A: A primitive root mod p is a number a such that a has order p-1. This means p-1 is the least positive…