Show that x2 = Y mod p has

Q: Find the first three values of the linear congruential generator X; = 2x;–1 +3 (mod 13) starting…

A:

Q: U(12) under multipilcation modulo 12

A: To find U(12) under multiplication modulo 12.

Q: Suppose that p, q are distinct prime numbers, prove that p9-1 + gp-1 = 1 mod pq

A: We use Fermat's Little Theorem, to prove our claim.

Q: Let x, y, z E Z and x = y mod n, where n ez+ Then: 1) Show that x - z = y -z mod n 2) Show that xz =…

A: The given question is about number theory. The solution is given below.

Q: Prove that if ca = cb (mod n), then a = b (mod n/d) , where d = gcd(c,n). %3D

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Q: Let p= 2k +1 be an odd prime number. Show that x* – 1 = (X – 1)(X – 2²) - · (X – k²) mod p.

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Q: Assume that k mod 15 4. Compute 11k mod 15. Show all your work clearly.

A: Topic:- basic maths

Q: Find the smallest integer a > 1 such that a –1 = 2a (mod 11

A: To find:

Q: 12. Prove that if gcd(a, n) { b, then ax = b (mod n) has no solutions.

A: To prove the given statement, it is equivalent to prove its contrapositive statement which is "if…

Q: Use Fermat's theorem, show that if ged(a, 155) = 1 then a121 = a (mod 155).

A:

Q: Prove that if ax = b (modn) then () x = () (mod ()) where d = gcd(a, n). Apply this result to 5x =…

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Q: For which positive integers a is the congruence ax4 ≡ 2 (mod 13) solvable?

A:

Q: Let p be a prime with p ≡ 3 (mod 4), and suppose that q = 2q + 1 is also prime. Determine if 2 is a…

A: let p be a prime with p≡3mod 4 and suppose that q=2p+1 is also prime determine if 2 is a square mod…

Q: Prove that the number of solutions to the congruence xk ≡ 1 mod p is gcd(k, p − 1)

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Q: 8, Suppose a,b € Z. Prove that a = (mod 5). b (mod 10) if and only if a = b (mod 2) and a = b

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Q: Show that if a ≡ b (mod m) and c ≡ d (mod m), then a + c ≡ b + d (mod m).

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Q: Prove that if x2 is congruent to y2 (mod n) and x is not congruent +or - y (mod n)  then gcd(x + y,…

A: Given: x2≡y2modn x≢±ymodn

Q: Find all x EZ221 such that x2 =97 (mod 221).

A: To find : all x∈ℤ221 such that x2≡97mod 221 Pre-requisite : Properties of Legendre symbol - 1. ap=1,…

Q: suppose a,b ∈ ℤ. Prove that a ≡ b (mod 10) if and only if a ≡ b (mod 2) and a ≡ b (mod 5).

A: In Solution we use divisibility formula that  If a and b are integers, a divides b if there is an…

Q: Find all integer values of x for which 3x = 4 (mod 8).

A: given congruence equation 3x≡4(mod 8)

Q: Let S := {1< a<p - 1: () = 1} . Ifp = 1 mod 4, show that Σ p(p – 1) 4 rES

A:

Q: (b) Prove that if a = 7 mod 10 and b = 6 mod 15, then a + b = 3 mod 5.

A: Our objective is to prove . Since you have posted multiple questions I am solving one.

Q: The quadratic congruence x^2+4x+1=0 mod 7 is solvable True False

A: Since not a particular question asked as per guidelines ,solution to only first question is given…

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

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

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Q: 1. Use Euler's Theorem to prove a 265 = a (mod 105) for all a E Z.

A: To prove:                   a265≡amod 105 for all a∈ℤ by using Euler's theorem.   Euler's theorem:…

Q: Find the smallest positive integer satisfying the system of congruences: x ≡ 1 (mod 3) x ≡ 2 (mod…

A: Here we will find out the required value.

Q: Show that y = x (mod p) leads to -y = -x (mod p).

A:

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: Show that: if a = b(mod n) and C = d\modn), then a +c =b+d\mod n)!

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

A:

Q: Find an integer x satisfying the two congruences: x = 1(mod 3) 2x = 0(mod 6), or show that no such…

A: The given congruences are                                 x≡1 (mod 3)2x≡0 (mod 6)

Q: 27 = x(mod 5) is true if x =

A:

Q: 3) If the gcd(z,n) = d and xz = yz mod n then: Show that x = y mod

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Q: Give an example to show that a ^2 ≡ b^2 (mod m) need not imply that a≡ b (mod m)

A: Here I listed given example of above statement.

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: Find all integers X satisfying X ≡ 1 mod 6, X ≡ 3 mod 8, X ≡ 7 mod 9

A: First divide 1 i.e. the dividend by 6 i.e. the divisor 1/6 = 0.16667 Then multiple the whole part of…

Q: 12. Prove that if gcd(a, n) b, then ax = b (mod n) has no solutions.

A:

Q: Prove that if a = b mod n and c= d mod n, then ac bd mod n.

A: Given if a≡bmodn and c≡dmodn We have to show that ac≡bdmodn. Now, a≡bmodn…

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: Prove that if p is a prime number then Z mod p is an integral domain

A: Integral domain:- A commutative ring R with a unit element 1 with no zero divisors is said to be…

Q: Show that if x mod 8 = 1, then (3x −5) mod 6 = 4.

A: Solution

Q: Show that if a = b mod m and b = c mod m, then we have a = c mod m. Use the definition of congruence…

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Q: (b) Solve the set of congruences 2x =1 (mod 5) x=3 (mod 4)

A: Note:- We’ll answer the first question according our protocol. Please submit a new question…

Q: Prove the above property. That is, prove that if a mod n = k and b mod n = j, then (a + b) mod n =…

A: If a mod n =k and b mod n=j show that a+b mod n =j+k mod n. Definition: If x mod y=z, then y | x-z.…

Q: Number theory

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Q: Show that if p is a prime number then xy ≡ 0 mod p implies that x ≡ 0 mod p or y ≡ 0 mod p

A:

Statements and Logic

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