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: 8. Solve the following congruences. a. 5x + 3 = 13 (mod 14) b. 4x = 20 (mod 8)

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Q: 2p a) n! = 2 (mod p), where for 0 <k < n. %3D k!(n - k)!

A: To prove the given part (a) 2pp≡2mod p where nk=n!k!n-k! for 0≤k≤n

Q: Given the following system (M). ( 2(mod 16) 1(mod 17D x= 3(mod 18) Nx = 2(mod 19) Which among the…

A: The given system is  x≡2mod16x≡1mod17x≡3mod18x≡2mod19

Q: I. Solve for the smallest value of x in the following linear congruences. 1. x = 3 mod 6 and x = 4…

A: To solve given linear congruences

Q: The linear congruence in two variables ax + by ≡ c (mod m), where a, b, c, and m are integers, m > 0…

A: As we know-the congruence ax+by≡c (mod m)has dm incongruent solution if   d=(a,b,c)|c&amp; no…

Q: Consider the congruence 4 a 2 (mod 3). Then the value of 'a' could be: O a. the solution does not…

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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: Q2: Solve the following (if possible): 1. = 5x + 11 mod 37 3

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

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

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: The equation 82r = 12 (mod 13) has O 1 incongruent solution modulo 12 O 4 incongruent solutions…

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Q: Solve each of the following congruences. (a) 6x + 3 = 4 mod 10 (b) 6x + 3 = 1 mod 10 (c) x² = 1 mod…

A: In congurent modulo n, ax≡bmod n has exactly 'd' solutions if 'd' divide 'b'where d=gcda,n otherwise…

Q: x is congruent to 27 mod 7. Then x is 54 55 None of the option 46

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Q: Furthermore, any two solutions x and y are congruent modulo n¡n2 · Solve the following systems of…

A: NOTE: Since you have asked multiple questions, we will solve the first question for you. If you want…

Q: 2. Prove directly if the statements that are true, give counterexamples to disprove those that are…

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Q: x is congruent to 31 mod 6. Then x is 40 54 55 None of the option

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Q: Q7. Solve the following set of simultaneous congruences. 4x = 1(mod 5) 6x = 5(mod 11) 2x = 5(mod 13)

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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: 5x = 7 (mod 32)

A: To solve the following problem

Q: Solve : ex+ay =4 (mod 13) E 2 2 (mod 13) Using elimination

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Q: 3. Solve each of the following congruences. (a) 6x + 3 = 4 mod 10 (b) 6x + 3 = 1 mod 10 (c) x² = 1…

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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: côngri 8.8. Use the technique of the previous problem to solve the following systems of congru-…

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Q: For which p= 3 mod 4 is i a square in F»[i]? For which p = 3 mod 4 is i the square of an element in…

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Q: 1. Find all solutions to the following congruences. (a) 2x +3 = 7 (mod 9) (b) 2x +3 = 7 (mod 20) (c)…

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Q: 1. Solve the following linear congruences using Diophantine equations: 25x = 7 (mod 17) and 26x = 8…

A: Given information: The Diophantine equations are 25x≡7mod17 and 26x≡8mod6. Calculation: The equation…

Q: 1. Solve the congruence 17x ≡ 42 (mod 100). 2.Compute the remainder of 3293 when divided by 97…

A: 1) solve the congruence 17x=42+100k implies17≡42+100k as we know gcd17,100=1 and 1/42 now, using…

Q: Which of the following are true modulo 12? (1) Does 5z = 6 mod 12 have a solution? How many? (2)…

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Q: (a) Let p E N be a prime number, p # 5, and let nE N. Determine the number of solutions to the…

A: First note that this question is only solvable for certain primes and not for all primes. We…

Q: Q7. Solve the following set of simultaneous congruences. 6x = 1(mod 5) 5x = 3(mod 11) 2x = 5(mod 13)

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Q: You and your friend Lara are considering the linear congruence ax = b (mod m), where m = 6 x 11 x…

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Q: -) Find the solutions, if any, of each congruence. 21x 10 (mod 35) 9x = 15 (mod 21).

A: We know that, ax≡b mod n has solution if and only if gcd(a,n) divides b.

Q: 24. Given the linear congruence (16x is congruent to 72 mod 24.) How many possible incongruent…

A: 24. To find the number of incongruent solutions of16x≡72mod 24

Q: Suppose b is any integer. If b mod 12 = 7, what is 5b mod 12? In other words, if division of b by 12…

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

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Q: which one will give a solution of [6,13] a)4x is identical to 3 mod 7 b)9x is identical to 11 mod…

A: which one will give a solution of [6,13]

Q: Solve the following congruence equations. 1. x=2 mod 3 2. 9x = 21 mod 30 3. 18x = 30 mod 42

A: "Since you have posted a question with multiple sub-parts, we will solve the first three sub-parts…

Q: 1. x=2 mod 3 2. 9x = 21 mod 30

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Q: 1. Solve the following linear congruences: (a) 25x = 15 (mod 29). (b) 5x = 2 (mod 26). © 6x = 15…

A: As per our guidelines we can answer only three subparts and rest can be reposted.  So, I am going to…

Q: Suppose b is any integer. If b mod 12 = 7, what is 6b mod 12? In other words, if division of b by 12…

A: To determine the value of 6b mod(12) if b mod(12) = 7.

Q: 4) The least positive integer residue of 21000000 mod 17 is a) - 1 b) 1 c) 2 d) a

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Q: A sequence of pseudorandom numbers is generated as follows Xo = 4 x₁ = ( 6x₁-1 + 5 ) mod 13 if i > 0…

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Q: mod 32, given Xo=1 Please answer w

A: The congruential generator is given as:Xi+1 = 11Xi mod 32, where we are given X0 = 1. We see that,…

Q: Deduce the following statements: i) an odd prime number p is the sum of two squares of integers if…

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Q: . Find a solution x € Z, 0 <x <n, for the congruence 57x + 7 = 78 (mod 53).

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Q: solution to the congruence equation (3x + 12) = 7 mod 10

A: Given :  3x+12=7mod 10⇒     3x=7-12 mod 10⇒     3x=-5 mod 10⇒     3x=5 mod 10        -5 mod 10 =5⇒…

Q: The equation 12x=24 (mod 48) has: O 24 non-congruent solutions modulo 48 6 non-congruent solutions…

A: To find- The number of solutions for the given equation.

Q: 10. Solve the following modular congruences or show that there is no solution. (^) 10r =1 (mod 31)…

A: We will find out the required value.

Q: 1. Find all solutions to the following congruences. (d) 5x = 10 (mod 20)

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Q: 3. Let d = (a, n) where n > 1. Prove that if there is a solution to ax = b (mod n), then d divides…

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