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: 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: s 19 a prime in the Eisenstein integers? is 79? If they are, explain why, if not, display a…

A: Given: Primes in Eisenstein integer. To find: a) To check wheater 19 and 79 are primes in Eisenstein…

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: 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: 2.4 Prove that if a and b are integers and p is a prime, then (a + b)² = ap+bP (mod p). You may…

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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 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: 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: Let be a rational prime, with p = 1(mod 4). Prove that p is the product of two Gaussian primes that…

A: We will use the basic knowledge of number theory to answer this question properly and completely.

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: 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: 1.21 Theorem. Let a natural number n be expressed in base 10 as  n=akak-1...a1a0 (Note that what we…

A: We know that 10 mod 3 = 1. So,

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: - 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…

Q: (a pour) = (d(1) a) +...+ d& T dG T dl) +04+ mos ad ppo desta sedang

A: Here we use that (p-i)^p=-(i^p) in mod p. We proof this by binomial expansion.

Q: Prove that for any prime, p = 1 (mod 4) there are exactly (p-1)/4 incongruent values of a 0 (mod p)…

A: By theorem (a,p)=1 then xn≡a modp has g.c.d(n,p-1) solution or no solution according as…

Q: 13. Prove that if n ≥ 2, then 2 - 1 is not divisible by n.

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Q: 7. Let p and q be primes and suppose that p = 2q + 1. Let h E F and assume that h 1. Pre that ² h…

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Q: Prove that for any odd prime p and a, b = Z, ap+bP = 0 (mod p) ⇒ ap+bP = 0 (mod p²).

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Q: Theorem 5.3. Let a be an odd integer. If a3 we have (29) a 2 = 1 (mod 2º), so there are no primitive…

A: Proof:                 We shall prove by induction on α. First we consider α=3. Let α be an odd…

Q: Exercise 6.3.4. Find all solutions modulo 10 to x² + 3x = 0 (mod 10). There are two distinct…

A: We employ the chinese remainder theorem to find the four solutions. Details typeset in latex in step…

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

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Q: Theorem 4.10. Let p be an odd prime. Then among 1, 2, ,p-1, exactly (p-1)/2 many are quadratic…

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Q: Let min EN 7 atZ be such theut Ca,m)=1, Ca,n)=| ¥ min)=\, Let K = Om (a) I j= On(a). 31K that ak=I…

A: Given that m,n∈ℕ and a∈ℤ with a,m=1, a,n=1 and m,n=1.  Suppose  k=Omaj=Ona With j|k To prove that…

Q: 2. Use Fermat's Little Theorem, or its corollary, to find the units digit of 72018 + 112019 + 132020…

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Q: e) Let p and q >7 be primes such that 2 +2P +1 =0 mod q. Prove that q = 1+6kp for ome natural number…

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

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

Q: ... 9 2 7. Suppose p is an odd prime and a ‡ 0(mod p). For each k = 1, 2, . define ek and rk by…

A: Given a and p. p is an odd prime and  such that . For each , define  and  such that , where .The…

Q: Let m 2 be any integer and m is not of the form 2, 4, pa, 2pa, i integer a with (a,m) = 1 we have…

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Q: Use a direct proof: p is a prime # and x ∈ Z. If x2≡x (mod p) then x ≡ 0 (mod p) or x ≡ 1 (mod p).…

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Q: II. For every odd prime q, define Mg = 29-1. Let p be a prime divisor of Mq. 1) Prove that ord,2 =…

A: We shall use part 1 to solve this problem. And we shall also use basic facts about primes. Further…

Q: Use the result in Problem 3 and state conditions on n, a, and k under which the congruence * a (mod…

A: For (4), Here, we need to find conditions on n, a, and k under which the congruence xk≡a mod n is…