(b) Give an example of a composite integer n such that the congruence x =1 (mod n) pas more than two solutions. Once again, here two solutions, x and y, are considered the same if = Y (mod n)

Q: What is the number of solutions to the congruence in Z125? x³ + x² + 3 = 0 (mod 125)

A: As per the question we have to solve the following polynomial congruence equation : x3 + x2 + 3 ≡ 0…

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: 6. Let p be an odd prime. Show that a2 = a (mod p*), k > 2 has exactly two solutions if x2 = a (mod…

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Q: TRUE or FALSE: 1. If a is congruent to b (mod m), then a^k is congruent to bk (mod m) for every…

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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: 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: 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: 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: (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: Exercise 37 Find an n where x² = -1 (mod n) has a solution and an n where x² = –1 (mod n) has no…

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Q: a) Prove that 10" =+1 (mod 11) for any n EN. b) Suppose the integer r has digits rI-1I, I0. Prove…

A: Want to prove:(a) 10n≡±1 (mod 11) for any n∈N(b) If the integer x has digits xnxn-1.....x1x0 thenx≡0…

Q: 24. Prove that for all integers m and n, if m mod 5 = 2 and n mod 5 = 1 then mn mod 5 = 2.

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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: Hello, I need help for the problem in the picture. Thank you!

A: We are given that α ∈ Z is a primitive root modulo an odd prime p. So,    αϕ(p) ≡ 1 mod p => αp-1…

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: (9) Find all prime numbers p for which both equations X²-2 0 mod p and have solutions. X²-3 0 mod p

A: Concept: The quadratic congruence: x2≡a(mod p) , where gcd(a,p)=1 Has a solution if (ap)=1 That is a…

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

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Q: Suppose n > 1 is a composite integer ab where a and b are unequal integers both greater than 1.…

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Q: 1. Find all simultaneous solutions of the congruences: (a) X= 3 mod 5 x= 4 mod 6

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Q: Let n ∈ Z. Prove that if n^2 + 2n −4 doesnt equal 1 (mod 5), then 5 doesnt divide n.

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Q: 3.1.12 Suppose that 7x = 28 (mod 42). By Theorem 3.9, it follows that x = 4 (mod 6). (a) Check this…

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

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Q: It is given that 3023 is prime, and 3023 = 2(1511) +1 where 1511 is also prime. (a) How many…

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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: True or False For arbitrary integers a and b, a ≡ b(mod n) if and only if a and b do not have the…

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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: We will prove the given statement.

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: Prove that if a1, a2, . . a, are n > 2 integers such that a; = 1 (mod 3) for every integer i (1 < i<…

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Q: You and your friend Lara are considering the linear congruence ax = b (mod m), where m = 6 x 11 x…

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Q: For every integer n &gt; 1, prove that (n - 1)!mod n = n - 1 if and only if n is prime.

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Q: Let 1Sasb be two positive integers. How can I prove or disprove that there exists a 1sks b-1 such…

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Q: Show that if p is an odd prime and a is an integer not divisible by p, then the congruence x2 ≡ a…

A: According to the given information, Let p is an odd prime and a is an integer not divisible by p…

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

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Q: A.)Many printed airline tickets contain a 10-digit document number followed by a check digit. The…

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Q: 1. Let p and q be distinct prime numbers and let a be an integer not divisible by p and not…

A: As per BARTLEBY GUIDELINES , I answered one question (i.e. Q-1) and repost other questions…

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

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Q: Suppose a and b are relatively prime positive integers. Is it true that (a+2b)4(ab) = 1 (mod ab) ?…

A: We try to solve the question using Euler's Theorem. The detailed solution typeset in LaTeX is…

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: Find all solutions to the congruence 10x≡5 mod 15 (b) Prove that 7 divides 32n+1+ 2n+2for any…

A: In  (a) we use the extended Euclidean algorithm to solve the problem. In (b) we use the mathematical…

Q: Prove that if n is an odd positive integer then 1 + 2 + 3 + · · · + (n − 1) ≡ 0 mod n . What happens…

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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: Theorem 11.3. If p and q = 2p+1 are primes, then either q | Mp or q |Mp +2, but presented next. not…

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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

Q: 1- Find a solution of 621m + 483n = k, where k is the gcd of 621 and 483.

A: 1. Given that, the equation is 621m+483n=k and k is the gcd of 621 and 483. Use the Euclidean…