(6) (a) Please determine 32049 mod 103 (Hint: 費馬小定理) (b) Suppose that x mod 43=2 and x mod 67=13 Please Determine x mod 2881. (Hint: Chinese Remainder Theorem) (c) n! = n(n-1)(n-2) .... 1. Please determine 39! mod 43 (Hint: Wilson's Theorem) (12 scores)

Q: Is there a solution to x^2 - 9x + 17 = 0 mod 73?

A: The given in this question is a quadratic equation in the form of x2-9x+17=0, and the objective is…

Q: let x  be a fixed but arbitrary integer. Do there exist infinity many integer y such that,…

A: Let x = x0 be fixed given integer. Then,

Q: x^2 ≡ 1 (mod 105)

A: Given: x2≡1(mod 105) To Solve: The given congruence  x2≡1(mod 105)

Q: (7) (a) Find the order of a modulo 7 for a = 1, 2, 3, 4, 5, 6.

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Q: Find the value of 592 (mod 11) using Fermat's Little Theorem. Remember, x (P-1) = 1 (mod p) a 592 =…

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Q: (7) (Z/50Z)*. Which of the other elements of (Z/50Z)* are also primitve roots modulo 50? Find a…

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Q: 1. How many solutions( mod 419) does x = 186 mod 419 have? (Show your work!) 2. How many solutions (…

A: Given: x2≡186 mod 101  To find: Number of solutions that exists. Method Used:  For any, xk≡b mod p,…

Q: Please answer True or False for the following Statements... 1. For every odd integer n, the integer…

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Q: Prove: If a and b are relatively prime integers, then a^(theta(b)) + b^(theta(a)) congruent 1(mod…

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Q: Y OU (a) Find the least residue of 639 modulo 10. (b) Use Fermat's little theorem to find the least…

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Q: Compute 17^7121 (mod 11).

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Q: Using appropriate methods, reduce (i) 419 mod 47, (ii) 141-10456 mod 47, (iii) 483 mod 47, (iv) (449…

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Q: a) For an odd prime p, prove that the quadratic residues of p are congruent mod p to the integers…

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Q: Using Fermat's little theorem, find the least residue of the multiplicative inverse of 5^53 - 4…

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Q: (b) By applying part (a), solve the congruences 2x = 1 (mod 31), 6x = 5 (mod 11) 3x = 17 (mod 29).…

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Q: 2. Find 52019 (mod 13).

A: Here,given  52019(mod 13)Now we find 52019(mod 13) using Chinese Remainder Theorem and Fermat's…

Q: If a≡3 (mod 11) and b≡9 (mod 11): (a) Compute a+b(mod 11) (b) Compute a^3(mod 11)

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Q: Prove the following using a direct proof.  If a,b, and c are integers such that b is a multiple of…

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Q: 75 (mod 46)

A: Take a number and modulo with 46

Q: Solve the polynomial congruence x^3 - 3x^2 + 27 = 0 (mod 225). Please be detailed! I’m really…

A: Solution are fully explained in below steps Explanation:

Q: How do I prove this: Show that if x ≡ 1(mod n), then x^2 ≡ 1(mod n) and x^3 ≡ 1(mod n) and how to I…

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Q: Use Hensel’s Lemma and Chinese Remainder Theorem to find all solutions to x^2 + 3x + 17 ≡0 (mod 300)

A: Introduction: Hensel’s Lemma is a powerful tool in number theory that allows us to lift solutions of…

Q: Compute (7^3)^161 (mod 250)

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Q: (d) 357 862 193 = (mod 943). (e) 5327 6135 7139 2187 5219 . 1873 = (mod 8157). (Hint. After each…

A: ...

Q: Show that (-3) = 1 if q = ±1 mod 12 -1 if q = ±5 mod 12

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Q: Need help with True or False for the following... (I will put what I believe As (T) or (F)) 1. For…

A: Let's evaluate the provided statements and their negations as well as the converses and…

Q: If n 2 1, then the Fermat number Fn 22" +1 is prime if and only if %3D 3(F.-1)/2 = -1 mod (Fn).

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Q: Work the modular arithmetic problem. (12 9)(mod 3). (12 9)(mod 3) =(mod 3)

A: The given modular arithmetic problem is -  (12·9)(mod 3) We have to solve the above problem.

Q: In the question that follows, the "Units Digit" or the "Unity Digit"of a number is the last digit in…

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