Consider the linear congruence 15 = 1113 (mod 10371). If given that gcd(15, 10371) = 3, then answer the following questions. (a) How many solutions does the congruence have? (b) Simplify the congruence. Show your work. (c) Find a solution to the simplified congruence you found in (b). Show ur steps. (d) Find the first two solutions.

Q: (a) Solve the linear congruence 26x = 1 modulo 33

A: Solution

Q: Consider the linear congruence 27 = 105 (mod 4965). If given that gcd(27, 4965) = 3, then answer the…

A: Notes: the variable x is missing in the question. I had rectified in my solution.

Q: Solve the quadratic congruence, 6x²-7x+2 ≡ 0 mod 31.

A: We have to solve the above quadratic congurence.

Q: Solve a congruence equation and give all the solutions: 3(x – 2) + 4x = 5x +8 (mod 31)

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Q: Solve the linear congruence 2370x ≡ 69 (mod 2517) or say why no solutions exist

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Q: Solve for : y = 14x + 4 (mod 51) Step 1. subtract 4 y- 4 154 Step 2. Find the multiplicative inverse…

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Q: Find all solutions of the system. 3x = 1 (mod 5) 2x = 3 (mod 7) 5x=11 (mod 3)

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Q: 4. Solve the following quadratic congruences: 5x² + 6x + 1 = 0 (mod 23)

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Q: Find all solutions of each congruence and explain why these are all the solutions. (It is possible…

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Q: For this question, explain the reason if no solution exists. Give a complete list of solutions if…

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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: Find the smallest positive integer solution to the following linear congruence: 7r 13 (mod 34).

A:

Q: (a) Solve for x if 25x = 15(mod 29).

A: Linear congruence

Q: For each of the following equations determine if they have a solution. If yes, find the solutions,…

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Q: D. Determine the number of incongruent solutions of each linear congruence. 1. 22x = 18 (mod 75)…

A: Given linear congruence is  1. 22x≡18mod 752.   2z≡119mod 983.   9x≡95mod 36

Q: (a) Find all solutions r to the congruence 341x = 16 mod 912

A: Since you have asked multiple question, we will solve the first question for you. If youwant any…

Q: Find the solutions of the congruence 15x2 + 19x ≡ 5 (mod 65)

A: The given equation is: 15x2+19x≡5(mod 65)

Q: (3) Solve the system of congruences: I = 2 mod 5 I = 3 mod 8 I = 7 mod 11

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Q: Solve the following linear congruences (you may if you wish, find some solutions by inspection): (a)…

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Q: x = 10 (mod 18), x = 14 (mod w). Solve system of congruences, find the smallest possible value for…

A: given x≡7mod 15   ...ix≡10mod 18   ...iix≡14mod ω   ...iii where ω≥20 to find solve system of…

Q: 3. Use congruences to prove the following: (a) 4"-1 is divisible by 3 for every natural number n.…

A: (a-b) is divisible by n.

Q: Solve each equation for x. (If there is no solution, enter NO SOLUTION.) (a)     x + 8 ≡ 1, (mod 9)…

A: We solve each equation for x.

Q: Consider the linear congruence 22 = 712 (mod 4366). If given that gcd(22, 4366) = 2, then answer the…

A: Given: Linear congruence 22x≡712 (mod 4366). For linear congruence ax≡b mod n has solution if gcd…

Q: Encrypt the message STOP using the RSA cryptosystem with key (2537, 13). Note that 2537 = 43 ⋅ 59, p…

A:

Q: 5. Solve the following quadratic congruence using the "completing the square" method: 9x² +66x +49 =…

A: 5. Given equation is 9x2+66x+49≡0(mod 79)   To Solve: Above quadratic congruence using the…

Q: k from problem 2 and the method used in the example at the end of solve the congruence: 7x³ = 6…

A: Given: Congruence is To Find : Solve the congruence equation

Q: How do I use the Chinese Remainder Thm to solve this system of congrunces below? x ≡ 1 (mod 3) x ≡ 4…

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Q: Solve the quadratic congruence x² = 35 (mod 817).

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Q: For the congruence 17x ≡ 9 (mod 276), write down an equivalent system of congruences. Hence solve…

A: We have to Write down an equivalent system of congruences and Hence solve the congruence 17x ≡ 9…

Q: Question 3 Using course notes, solve the congruence x³-7x+6=0(mod3².7²)

A:

Q: Find a Diophantine equation equivalent to 3x = 5 (mod 11) and use the solution of the Diophantine…

A: Congruence : Let us consider the equation a ≡ b (mod m)  here on dividing a by m the remainder will…

Q: 2) Solve the following linear congruences: 14x = 91 (mod 189) 24x = 56 (mod 88)

A:

Q: If possible, find a complete set of mutually incongruent solutions of each of the following…

A: The given congruences, a) 9x≡12mod 15. b) 81x≡57mod 117. We have to find the solution to the…

Q: Consider the linear congruence 22 = 712 (mod 4366). If given that ged(22, 4366) = 2, then answer the…

A: consider the linear congruence 22≡712mod 4366 given that gcd22, 4366=2 (a) how many solutions does…

Q: Using course notes, solve the congruence x³ + 3x² - 4 = 0(mod11²3²)

A: As per the question we are given the following congruence equation :x3 + 3x2 - 4 ≡ 0 mod 112×32…

Q: Use the index system modulo 19 to find all solutions of x³ = 11 (mod 19) and 5ª = 4 (mod 19)

A: In the theory of congruence, Fermat's little theorem is very significant. It says that for any prime…