9. Determine which of the following congruences have solutions, and find any solutions (between 0 and the modulus) by trial and error. a. 4r = 5 (mod 7). b. 67 = 4 (mod 9). c. r' = 2 (mod 7).

Q: Solve the following Diophantine equation. 25x + 13y = 2

A: In this question, the concept of the Diophantine equation is applied. Diophantine equation A…

Q: (1) Determine the number Sm of solutions to the equation xyz = 1 (mod m) for each positive integer…

A: Given that:To find:The number  of solutions to the given equation for each positive integer…

Q: Which of the following values are needed to compute 406 (mod 37) using fast exponentiation? Mark Y/N…

A:

Q: Given n=35. x=13. Find out the value of y=? (all possible) such that (x*y) mod 35=1.?

A: We will use (13y)mod35=1 to find all possible values of y by writing given congruence in expanded…

Q: 2. Solve the Root Mod x = 3 (mod 11) xE [5]10 OxE [7]i0 none of these xE [7]1 Oxe[5]1

A: Use Farmet's theorem

Q: II. Give FIVE different answers for each of the following. 1. 49 = mod 10 2. 82 = mod 9 3. 100 = mod…

A: 1) Given expression is  49≡______ mod 10

Q: Show that x2 = Y mod p has

A:

Q: 36. Find all solutions to each of the following linear equations: a. 3x= 4 (mod 5) b. 4x = 4 (mod 6)…

A: a)  In modular arithmetic, you can find the solution by multiplying both sides of the congruence by…

Q: 16. Which of the following congruencies has NO solution? . A. 24x = 18(mod 39) B. 5x = 4(mod 10) C.…

A:

Q: 1. Solve each of the folling. а. с. d. 115x = 2 (mod 288) 2. Explain why 12x = 1 (mod 75) has no…

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Q: 1. Solve each of the following. 14x = 1 (mod 45) 24x = 35 (mod 59) а. b. 62x = 1 (mod 99) С. d.

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Q: Consider 64z 16 mod 196. The number of solutions mod 196 is Oa None Ob 3 Oe 1

A: We find the number of solutions incongruent mod 196

Q: Solve the congruence 11x = 45 (mod 97) where x is an integer in the interval [0,97).

A:

Q: 1. Find all r €Z satisfying each of the following equations. (a) 3r = 2 (mod 7) (d) 9r = 3 (mod 5)…

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Q: Which of the following is TRUE? -52%3D0( mod 13) OA. -1=-13( mod 13) OB. 446=-278( mod 7) OC. 1=0 (…

A: Only (A) is true.  The detailed solution is as follows below:

Q: Solve for x. All final answers must be in Zn where n is the modulus. x≡−12 mod 13

A:

Q: 5. Let f(x) = 2x² + x + 4. It is known that f(1) = 0(mod 7). (i). Find a solution to f(x) = 0(mod…

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Q: (d) 9r = 3 (mod 5) (e) 5x = 1 (mod 6) (f) 3x = 1 (mod 6)

A:

Q: (3) Solve for x (give your answer as an integer between 0 and 7: 14 · 51 ≡ x(mod7)

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Q: 1. Let p be prime and let a 0 (mod p). Let b = a 2 (mod p). Show that ab = 1 (mod p). 2. Use the…

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Q: 5. Prove or disprove the following proposition: Let the Jacobi symbol: =1. %3D Then a is quadratic…

A: The complete solutions are given below

Q: (a) Prove the following: i. 11 + (6 x 4") - 22n = 6 mod 7, 1 is divisible by 131, ii. 8260 iii.…

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Q: 3. Find all solutions to x² + 5 = 0 (mod 245).

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Q: Solve for x. All final answers must be in Zn where n is the modulus. 6x mod 11 ≡1

A: Solving this is equivalent to finding multiplicative inverse of 6 in  Z11 . We have to find x such…

Q: Find d

A: The given question involves the concept of modular arithmetic and the use of the modular inverse.…

Q: Use calculus to sketch the graph of the given function. (a) f(x) = 3x* – 4xr

A: Consider the given function, fx=3x4-4x3 Here we need to sketch the graph of the given function using…

Q: Solve for x. All final answers must be in Zn where n is the modulus. x≡3(15) + 45 mod 18

A: The given equation is x≡315+45 mod 18 To find: the value of x.

Q: One of the following is a solution of 81x = 54 (mod 108) 43 47 45 42 has no solution

A: We are given the Equation :

Q: Solve each of the following congruences. Make sure that the number you enter is in the range [0,…

A: (b) 58x = 130 (mod 182) Let us consider:  58x = 182t + 130 Substitute the value of t = 0, 1, 2, 3, .…

Q: Solve for x. All final answers must be in Znwhere n is the modulus.   x+ 2 mod 9 ≡ 0

A: We have given that ,  x + 2 mod 9 ≡ 0 We need to find the value of x under modulus 9. We know that…

Q: 8.5. Find all incongruent solutions to each of the following linear congruences. (a) 8x = 6 (mod 14)

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Q: The number of mutually incongruent solution(s) of 306x = 90 (mod 360) 20 30 18 O 6 0

A: