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 two solutions for all a solution. Then prove positive integers k

Q: Let p be a prime and k a positive integer such that ak mod p = a mod p for all integers a. Prove…

A: Consider the provided that, ak mod p=a mod p,                ....1       where p is prime and k is a…

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: Let n = pq, where p and q are distinct primes such that p = q = 3 (mod 4). Suppose a E Z+ with a <…

A: We use the properties and definitions of Jacobi symbols to solve this problem. The detailed solution…

Q: (a). Find all possible integers k such that the following linear congruence has a solution. (9k? +3k…

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Q: Prove that : If p is a prime number and p does not divide a then a- 1(modp).

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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: For p an odd prime in Z , Prove: If p=(a)^2 + (b)^2  is possible in Z then   p= 1 (mod 4)

A: Compute general prime that satisfies given condition as follows.

Q: Let p be an odd prime number and an integer ga primitive root with respect to mod p. So is the…

A: We will use basic knowledge of elementary number theory to solve this question.

Q: (a) Find three distinct integers satisfying the equation r = 1 mod 11. (b) Prove that if k is an…

A: a≡bmod m means a-b is divisible by m where a and b are integers. For example 13≡3 mod 5⇒13 - 3 is…

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: Let p be a odd prime number and b, c € Z. Assume that there exists l e Z, such that l² = b² – 4c mod…

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Q: 1. Let n = pq, where p and q are distinct odd primes. Define X(n)= (p-1)(q-1) gcd(p-1,q-1)* Suppose…

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Q: Show that: Ifp is odd prime of the form 4k+1, then find a solution of x +1 0 mod p

A: We shall use Euler’s Criterion here. Euler’s criterion states as follows: “Let p be a prime. The…

Q: Let n and a be positive integers and let d = gcd(a, n). Show that the equation ax mod n = 1 has a…

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Q: Let n be a positive integer, and a an integer Prove that if the order of a modulon is n-1 then n is…

A: The given problem is to prove that n is prime from the given problem given a modulo n is set of…

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: Let n be a positive integer, and a an integer Prove that if the order of a modulo n is n-1 then n is…

A: Problem 2.  Solving it using the concept of phi function

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: Number theory

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Q: Suppose that p is a prime such that p = 5 (mod 8). Let k E Z such that p = 8k+5. For any a E Zp,…

A: 1. Let a∈ℝn. Then a is called quadratic residue modulo n if there exists an integer x such that…

Q: Establish that if a is an odd integer, then 2" aʻ = 1 (mod 2" *2) for any n 2 1.

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Q: 2. Let p be a prime number such that p> 5. Prove that p2-1 =0 (mod 24).

A: To prove the below statement If p is a prime number and p &gt; 5. p2−1≡0mod24

Q: Establish that if a is an odd integer, then for any n ≥ 1 a 2 n ≡ 1 (mod 2n+2). [Hint: Proceed…

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Q: Let p = 2^k + 1 be a prime for some k ≥ 1. Let a be any integer such that gcd(a, p) = 1 and a is NOT…

A: Consider the given information:

Q: 7. Let p be a prime number and n= 2P – 1. Prove that 2"-1 = 1 (mod n).

A: Let p be a prime number and n=2p-1. Prove that 2n-1≡1 (mod n)

Q: Find all positive primes p such that n = 1001p satisfies that a" = a mod n for all a.

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Q: Given any odd prime p, prove that: 2(p−3)! ≡−1 mod p. (Hint: Start with 2(p−3)!≡x mod p. Then work…

A: Let p be any odd prime number.  So we start with  2p-3!≡x mod p Also we have p-2≡-2 mod pp-1≡-1 mod…

Q: (5a) Let A = {n E Z | n = 2 (mod 3)} and B = {n E Z | n = 1 (mod 2)}. Prove that if n E (A n B),…

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Q: Prove that if gcd(a, n) = 1 and gcd(a – 1, n) = 1, then 1+ a + a² +a³ + ..+ a®(n)-1 = 0 (mod n).

A: we will prove that gcd (a, n) = 1, which  means that a and n are  prime. Let m= gcd (a, n) = gcd (a…

Q: 9. Let p be an odd prime, and let ( - 1)(a – 2).. (a - (p - 1)) = "--g12-2 -3 -...+ ap-32 - p-2a +…

A: Let a be any solution of x^2 + 1 ≡ 0 (mod p), so that a^2 ≡ −1 (mod p). Becausep doesn't divide a.…

Q: (a) Let p be an odd prime. Give an analogue of the quadratic formula mod p, and prove it. (b)…

A: (a) Given that p is an odd prime.  The quadratic formula mod p is given by: ax2+bx+c≡0mod p Here, p…

Q: If p and q are distinct primes and a is an integer such that aº = a(modq) and aª = a(modp), then

A: This is a problem of theory of congruence, number theory.

Q: Let a be an integer and let n e N. (a) Prove that if a = 0 (mod n), then n | a.

A: By the definition if congruence let a, b and n are integer, with n &gt;0, then a is congruent to b…

Q: Find the smallest positive integer k satisfying the stated condition or explain no such k exists:…

A: To find the smallest positive integer k satisfying the given condition of congruence.

Q: show that If n is a positive integer and a is any integer prime to n, then a ln) = 1(mod n), where…

A: It's called Euler's generalization to Fermats theorem. I have used integer m instead of n.

Q: Establish that if a is an odd integer, then 2" a“ = 1 (mod 2" *2) for any n 2 1.

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Q: Prove that if n is an odd positive integer, then n² = 1 (mod 8).

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Q: 4) The least positive integer residue of 21000000 mod 17 is a) - 1 b) 1 c) 2 d) a

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Q: Let p >2 be a prime. Prove that [(p-2) = (-1)/2 (mod p).

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Q: Show that for x and y integers, and m a positive integer x = y mod m implies x = y mod m.

A: This is a problem of Number Theory, Modular Arithmatic.

Q: Establish that if a is an odd integer, then 2" = 1 (mod 2" +2) for any n 2 1.

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Q: Prove: for any integer x, if x mod 3 is not 0, then (x2−3x+ 1) mod 3 = 2.

A: To prove :     for any integer x, if xmod 3≢0, then x2-3x+1mod 3=2   Pre-requisite : 1. a≡bmod n…

Q: Q2/let n be a fixed positive integer and a,b be arbitrary integers. Then a = b (modn) iffa and b…

A: We have to prove that Given statement

Q: If p is a prime satisfying n < p < 2n, show that 2n = 0 (mod p)

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Q: 2.1.3. If a, b are integers such that a = b (mod p) for every positive prime p, prove that a = b.

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Q: For n=195, by Fermat's theorem d-2 a) a (modn) b) 1(modn) c) 0(mod n) d) a(mod n)

A: Given:                 n=195 To find:                an-2=? by using Fermat's theorem Fermat's…

Q: For n=195, by Fermat's theorem a-2 a) a' (modn) b) 1(modn) c) 0(modn) d) a(mod n)

A: Fermat's theorem

Q: 6. Show that if n (a2" - 1)/(a² - 1), where a is an integer, a > 1, and p is an odd prime not…

A: The given question is related with pseudoprimes. We have to show that if n = a2p - 1a2 - 1, where a…

Q: Let n > 2 be a positive integer and r a primitive root of n. Suppose that a E Z is such that ar? 1…

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