Prove that the number log29 47 is irrational.

Q: xplain why F, contains a primitive nth root of unity if and only if p = 1 mod n. int: Use the…

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Q: What are fermat numbers? What is the first fermat number?

A: In mathematics, a Fermat number, named after Pierre de Fermat, who first studied them, is a positive…

Q: How can I prove that if x is a positive irrational number, then 2x-1 is also irrational

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Q: Solve xb =9 mod11

A: Solution:        As 11 is a prime number, by fermat's theorem,        x11-1 = 1 (mod 11)      =>…

Q: Show that log, q is irrational for any distinct prime numbers p and q. (Distinct here p# q.) means

A: To Determine :-  logp q is irrational for any distinct prime numbers p and q.

Q: Find all solutions for x, up to congruence.  x ≡ 2^63 (mod 61).

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Q: We know that x^2= 2(mod 112) and try to find the solution of x

A: This is a problem of Number Theory.

Q: Using Euler's Theorem, Find the last two digits of 123402

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Q: Solve y² = 37(mod 77)

A: We have to find the value of y  Using quadratic reciprocity property we find value of y

Q: 2. By using Fermat's Theorem, solve, 21x = 12(mod 17).

A: First, we simplify the equation by reducing the coefficients modulo 17. This is because in modular…

Q: Let n be an odd integer, a > 0, and b < 0. Prove that x" + ax + b= 0 has a unique root. Hint: If…

A: Let n be an odd integer, a &gt; 0, and b &lt; 0. Prove that xn + ax + b = 0 has a unique root

Q: Find all four solutions to x2 =6 (mod 19 x×23) with 1<x< 437.

A: Solution:-

Q: If p times tne pth term of an A.P. is equal to q times the qth term of the same AP., then how that…

A: Given that If p times the pth term of an A. P. is equal to q times the qth term of same A. P.  To…

Q: Find the primitive roots of following using discrete log a)log2 5 mod 15 b) log5 25 mod 29

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Q: Fill in the following blanks. Suppose you are finding 15^-1 (mod 131). This problem is equivalent to…

A: Answer follows

Q: The value of E[(i=O)^4]i! is О 32 O 36 O 30 O 34

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Q: From the statements below select the correct ones. A function f from A to B is onto iff for every x…

A: This is a question of function.

Q: 3) The number 2 is a primitive root modulo 19. Compute the powers 2' for I = 1,2, ..., 18, 19 to…

A: Given: Let us consider Modulo 19, the number 2 is a primitive root. To get the table for the…

Q: 2.23 Find all four solutions to the equation x²10 (mod 35).

A: The given equation is: x2-1≡0 mod 35 We have to find all four solutions to the given equation.

Q: Question 2. Prove that v2 + v2 is irrational. [Hint. Calculate the first 6 powers of this number.]

A: We will use the rational roots theorem to prove that 2+23 is an irrational number. The rational…

Q: 3x = 11 (mod 20)

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Q: Evaluate |1 – cos40 de O 1/(2v2) V2/2 O None of the choices O 1/2

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Q: 6 Show that for an integer n ≥ 2, the period of the decimal expression for the rational number is at…

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Q: 10. Let x be a real number. Prove that at least one of the numbers /2 – x or /2 +x is irrational.

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Q: 9. Prove or disprove: The inequality 2 ≥ x + 1 is true for all positive real numbers x.

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Q: d (sec-¹(2)) is equal to O O 1 2 √√4-1 1 In 2√4-1 O 1 √4¹-1 In 2 √4-1 In 2 2√4-1

A: Solution: ddxsec-1(2x) Apply the chain rule : df(u)dx=dfdu·dudx f=sec-1(u), u=2x =ddusec-1(u)ddx(2x)

Q: Suppose that m andn are positive integers and b is a real number so that the principal n" root of b…

A: We have been given that nth root of b exists where b is the real number.

Q: Solve the equation 5x = 13 (mod 18)

A: Given: To solve: the equation 5x4≡13mod 18Formula: a≡bmod cis equivalent to saying by Euclid's…

Q: where _13-4 mod 13

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Q: 0, the remainder is 2; and when divided by 11, the remainder is 0. remain is 5; when divided by

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Q: Show that given any particular real number, such as I, there is NO next real number immediately…

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Q: Use the discrete logarithm modulo 11 with base 2 to solve the following congruences: 4x2 = 9 mod 11

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Q: 10 Solve 3Y=2 (mod 7) 8 Salutions exist or show that

A: We have to solve given problem:

Q: Exercise 1.2.2. Show that there is no rational number r satisfying 2r = 3.

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Q: Find the modulus and argument of 5 - 7i.

A: Since the modulus of a complex number a+ib is |a+ib|=a2+b2. So, the modulus of 5-7i is…

Q: With fn the nth Fibonacci number, show that fn+1fn-1 – få = (-1)" whenever n > 0.

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Q: 5. Prove that the following are irrational: E-2-3', £-12¬!.

A: Expand the given functions and then determine their values. If the values are irrational then the…