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## Related Advanced Math Q&A

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Q: Show that th equation x" + y² = z" (mod 5) has nonzero solutions in Zz if and only if n is odd.

A: Given: xn+y2n≡zn(mod 5)

Q: 2. Find the largest negative x € Z that simultancously satisfies 4x = 9 (mod 5) 2x = 0 (mod 6) x =0…

A: Given : 4x = 9 (mod 5) 2x = 0 (mod 6) x = 0 (mod 4)

Q: 6. Let Fn 22" + 1, n > 1, be a prime Fermat number. Show that 3 is a primitive root modulo Fn.

A: Given: Fn=22n+1,n>1

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: 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: 11- P, X Pz X P, ... X Pn, where M= Pr. Pn are distind primes Prove by inducton, For all Ppositive…

A: Fermat's little theorem is one of the most important and significant methods in number theory.…

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: Show that 6n – 1 and 12n+1 are relatively prime for any n e Z.

A: Click to see the answer

Q: Suppose n is an integer and consider the number n* – 6n³ – 18n² + 6n + 1. a) By expanding the…

A: Explanation: a. Given that, Expand the right side and show that both sides are…

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Q: 4. Find d, the gcd (2695, 1547). Hence or otherwise, express in the form 2695p + 1547q = d, p, q E Z…

A: Click to see the answer

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Q: 4. Find d, the gcd (2695, 1547). Hence or otherwise, express in the form 2695p + 1547q = d, p,q E Z…

A: 2695 = 1547(1) + 1148 1547 = 1148(1)+399 1148 = 399(2) + 350 399 = 350(1) + 49 350 = 49(7) + 7…

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Q: 6. Let f(x) be a polynomial with integral coefficients (like f(x) = 3x^^5-x^2+7) and suppose there…

A: Given:- fx=3x5-x2+7 There are 2 solutions x mod 5 such that fx is congruent to 3 mod 5 and there…

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Q: k) Prove that x = 5108 = 179 (mod 433) is a solution to the equation x² + 1 = 0 (mod 433), and use…

A: 52 ≡25(mod 433)54 ≡625≡192(mod 433)55 ≡960≡94(mod 433)56 ≡470≡37(mod 433)512 ≡1369≡70(mod 433)524…

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Q: п 6. Prove that E(8i – 5) = 4n² -n for every positive integer n. п i=1

A: # Given: sum(8i-5) =4n^2-n i=1,2,3,.....n We have to prove the above equation??

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Q: Does x = a (mod p) always have a solution for every value of a, whenever p is prime?

A: Click to see the answer

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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…

A: Click to see the answer

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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),…

A: Click to see the answer

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

A: Click to see the answer

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Q: (b) Give an example of a composite integer n such that the congruence x =1 (mod n) pas more than two…

A: Click to see the answer

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Q: 2X+3=1 mod 5 Show your work

A: We have solve given problem:

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Q: (i) Prove that if p is a prime, then (p - 1)! = -1 (mod p). (ii ) Show that 18! = -1 (mod 437).

A: Click to see the answer

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Q: Number theory

A: Consider the provided question,

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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…

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Q: 8. Find the remainder of x2021 + x2020 + 1 modulo x2 – 1 in Q[x].

A: Remainder of the Polynomial

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Q: 1. Prove that if 7n2 + 5 is even, then n is odd

A: The given statement is,

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Q: For which primes p > 5 is -5 a square mod p? Your answer should be of the formp = {a1, a2, ..., ak}…

A: Click to see the answer

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Q: Find the first few terms of the sequence of pseudorandom numbers generated using the linear…

A: Given below an step by step solution

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Q: n>1 Problem 4. Prove by induction that, for every geq2 1 n +1 - - - 32 n2 2n

A: Click to see the answer

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Q: Find the smallest integer value c and it's n, to show that 2n2 + 6n -3 Oln² –10) Is an

A: Click to see the answer

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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.

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

A: Solution: If n is not a prime number and gcda,n=1 then by the Fermat's theorem aϕn≡1mod n where, ϕ…

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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…

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Q: Find the first few terms of the sequence of pseudorandom numbers generated using the linear…

A: Click to see the answer

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Q: 8. Solve the following quadratic equation via the two different methods, namely both classical and…

A: Click to see the answer

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Q: Let be a prime number. Stating any results that you use, deduce that xº-1 = (x – 1)(x – 2) · . · (x…

A: Click to see the answer

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Q: 1. For In 5n2 + 9n3 + 1, ɑn =n², prove or disprove: In O(an) (n → ∞)?

A: Given: xn =5n2 + 9n3 + 1, αn=n2. we have to prove or disprove : xn =O(αn ) ( n→∞).

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Q: 4. The least positive residue of 2^1000000 mod 17 is: * 1 None of these

A: Question 01: By Division algorithm, 10,00,000= 16*62,500 and since 17 is prime and 17 is not…

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Q: 5. Prove that the equation ø(n) = ¢(n+ 2) is satisfied by n = 2(2p – 1) whenever p and 2p – 1 are…

A: Euler totient function formula: p is prime then ϕ(p)=p-1 ϕ(pk)=pk-pk-1 ∀k∈ℕ If gcd(a,b)=1…

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Q: Find the first few terms of the sequence of pseudorandom numbers generated using the linear…

A: Click to see the answer

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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…

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Q: If 2n + 1 is prime for some n ≥ 1, prove that n is a power of 2.

A: We have given 2n +1 is prime for some n ≥ 1 We have to prove that n is a power of 2.

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Q: |State whether the following statements are true or false. Show working. a. For each odd prime p, a…

A: Click to see the answer

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Q: Recall the Fibonacci numbers Fn are defined by Fo = 0 , F1 = 1 and Fn Fn-1 + Fn-2 for all n > 2 .…

A: Consider the provided question, Fibonacci numbers Fn are defined by F0=0, F1=1 and Fn= Fn-1+Fn-2 for…

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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…

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Q: 1 (b) Prove by induction that (3i – 2)² = ;n(6n² – 3n – 1) for n 2 1

A: We have to prove ∑i=1n3i-22=12n6n2-3n-1 for n≥1 by using mathematical induction.

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Q: Suppose b is any integer. If b mod 12 = 5, what is 6b mod 12? In other words, if division of b by 12…

A: Given b is an integer such that b mod12=5 To find the value of 6b mod12

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Q: What is the smallest integer that is a quadratic residue for all prime moduli p > 2?

A: We know if there is an integer 0<x<p, where p is a prime, such that x2≡qmod p, then q is said…

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Q: 3V Show that there dã do not exist two real numnbers which add to 7 and multiply to 13. It is not…

A: our objective is to conclude the result

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Q: Let a, b, and c be integers and n a natural number. If a ≡ b mod n then a+c ≡ b+c mod n. How do I…

A: Click to see the answer

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Q: If |G| = pq, where p and q are primes that are not necessarily distinct,prove that |Z(G)| = 1 or pq.

A: Given: G=p.q where p & q are primes To prove: ZG=1 or pq