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). Show all work clearly and neatly. Do not skip steps.

Q: Prove that there are infinitely many primes congruent to 3 mod 8. previous part.

A: We have to show that there are infinitely many primes congruent to 3 mod 8. we will proof this by…

Q: Prove this claim: For every positive integer d and every integer n, the following equation is…

A: Using division algorithm we solve this problem. The steps are given bellow.

Q: Prove that if a = b (modn) and m/n, then a = b (modm).

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Q: Use trial and error to find all replacements for x that are less than the modulus that make the…

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Q: If a = b (mod n) and a = c (mod n) then which of the following is true. (a) b = c (mod n) (b) b2 =…

A: We use the properties of congruences to answer the given question.

Q: Let a and b be integers satisfying a=3(mod 11) and b=8 (mod 11). If c=3a+4b then None of these c=9…

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Q: 2. If a = 1 (mod p), then a=1 (mod p). Select one: True False

A: If , then . The truthfulness of the given statement can be proved by taking an arbitrary value for…

Q: True or False : a – c ≡ b – c (mod n) ⟺ a ≡ b (mod n)

A: We have to solve given problem:

Q: 5- If a = b mod(n), then ac = bc mod(n). 6- If a b mod (n), then ak = b*mod(n) for every positive…

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Q: Show that 127 = -16(mod 13) and show that 300 = 1(mod 13).

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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: Can you please show me step by step for this math that makes me undertandable. Q: Determine which…

A: (a) For the congruence 4x ≡ 5 (mod 6), we need to find a solution for x such that when 4x is…

Q: Assuming Wilson's Theorem, prove that if p prime and (a, p) = 1, then aP-1 = 1 mod p.

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Q: (a) Show that, for any odd prime p > 5, 1 if p = 1, –5 (mod 12) (주) -1 if p= -1,5 (mod 12)

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Q: (d) (e) (f) Assume that a 15 = 1 Find the value of ord41a. mod 41 and a #1 mod 41. How many…

A: As per the question we have to find : ord41(a) , where a15 ≡ 1 mod 41 but a ≠ 1 mod 41 total number…

Q: Suppose that p is a prime with p = 5 mod 8 . Show that the congruence px2 + 6y = 1 mod 3p has no…

A: Consider the given information.

Q: Compute (2100 mod 3), (2100 mod 5), and (2100 mod 7). Explain how you got the answers.

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Q: Let (a,b)=1 with a = c (mod b) and bª = c (mod a) for some integer c. Prove that ab+(b). +ba+(a) = c…

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Q: True or False: 5 is a primitive root modulo 29. O True O False

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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: Select the following statements that are true. 261000 = 1 (mod 3) (2022) 10 (FC6) 16 In Z6, 2¹⁰ = 2…

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Q: (a) Prove that 318 = q (mod 19).

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Q: Podasip 11. If p is prime, then every c can be expressed as a sum of two squares: x² + y² = c(mod…

A: The given is statement is not accurate. The correct statement in connection with the given statement…

Q: Hanb(mod n) and b c(mod n) then which of the following is trae. (a) anc (mod n) () 2=c(mod n) (c)…

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Q: True or False. Briefly explain why.  By Fermat’s Little Theorem, 6 is the smallest positive integer…

A: The given statement is FALSE. Fermat's Little Theorem states that if p is a prime number and a is an…

Q: Suppose that a and b are integers, c b = 3 (mod 19). Find the integer c that a) с%3D 13а (mod 19).…

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Q: proof that if a equiv b(mod n), then a+c equiv b+c (mod n) and ac equiv bc (mod n).

A: Apply the definition of congruence and divisibility, to express a≡bmod n. From the definition of…

Q: If a+c=b+c (mod m) then a =b (mod m) Select one: True False

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Q: 20.What is the value of n such that (24  is congruent to 58 mod n) ? a.13 b.17 c.-34 d.-13

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

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

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Q: (d) For all integers a and b, if ab = 7 (mod 12), then either a = 1 (mod 12) or a = 7 (mod 12).

A: (d) For all integers a and b, if ab=7 mod 12, then either a =1 mod 12 or a =7 mod 12

Q: (32³ mod 13)² mod 11

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Q: 1. Let a and b be natural numbers, where a, b > 2, such that gcd(a, b) = 1 and ab = (x + y)" for…

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Q: Assume all variable as positive integers. Given and a-2n+2 (mod 7) 000 b = 3n+1 (mod 7) Which of the…

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Q: (f) If a = b (mod n), then a* bk (mod n) for any positive integer k.

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Q: In the following question: " Let a, b ∈ Z. Show that if a ≡ 5 (mod 6) and b ≡ 3 (mod 4), then 4a +…

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Q: Let x, y ∈Z. Prove that if x≡1(mod 5) and y ≡2(mod5 ), then x^2-y^2≡0(mod 5).  Give 2 examples to…

A: Given, x,y∈ℤ and x≡1mod 5, y≡2mod 5

Q: (14.)(a) Prove or disprove: If ab= 0 (mod n), then a 0 (mod n) or b = 0 (mod n). (b) Do part (a)…

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Q: qual to 2.

A: In this question, concept of Modular Arithmetic is applied. Modular Arithmetic Modular arithmetic is…