Solve the following congruences. Make sure that the number you enter is ≥0 and ≤N−1, where N is the modulus of the congruence. If there is more than one solution, enter the answer as a list separated by commas. X1−215≡742 (mod74) X1=  X2+712≡166 (mod43) X2= (X3)^2≡9 (mod13) X3= (X4)2≡1 (mod8) X4=

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

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Q: Use this theorem to solve the following congruences: x1 = 13 mod 35

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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& no…

Q: 2. Solve the following polynomial congruences. ces (a) x² + 12x – 17 = 0 (mod 143) (b) x³ + 30x² +…

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Q: For each of the following primes, find the smallest positive integer k such that k is a prim root…

A: A primitive root of a prime p is an integer g such that g(modp) has multiplicative order p-1.

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: 3.1.4 Find all integers x which satisfy the congruence equation 3x = 2 mod 8.

A: 3x = 2 (mod 8) So, 3•(3x) = 3•2 (mod 8) So, 9x = 6 (mod 8) So, x + 8x = 6 (mod 8)

Q: Find all incongruent solutions - if exist - for the congruence: 36r = 96 mod 156

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

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Q: 2.x = 5 (mod 3) 5x + 4 = 5 (mod 7) d.

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Q: Let n and a be positive integers and let d = gcd(a, n). Show that the equation ax mod n = 1 has a…

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Q: {-74 + 1, ..., 74 – 1}. (b) Hence, find a solution to the congruence x' = 99 mod 74.

A: Given:- An expansion above Polynomial will have co-efficient lies between {-74+1,..........,74+1}

Q: Given the following linear congruence (H): 2x+1=-10 (mod9). Which among the following is true? O (M)…

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Q: 1) Find a solution – if possible – for each of the following two congruences. If for one of the…

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Q: The linear congruence 7x = 3(mod16) is solvable of solution: O None of the choices

A: The given problem is to solve the linear congreunce equation, we have to use congruence properties…

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: Given a = 4( mod 9) and b = 5( mod 9), find c with 0<c<9 such that c = 5a + 3b ( mod 9)

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Q: Find all solutions to the following quadratic congruences by first making the leading coefficient 1…

A: Topic:- algebra

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: Find all non-congruent solutions to the following congruences. Express solutions using the standard…

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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: 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: If a and b are positive integers and p is prime with p ł a, then the solutions of the linear…

A: We will check whether the given statement is true or false.

Q: Use Hensel's Lemma to find a solution to the following congruence: a +3 x + 3 = 0 mod 54.

A: Let f(x) be a polynomial , p is a prime number . If x=a is a solution of congruence equation…

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: Find all incongruent solutions - if exist - for the congruence: 36x = 96 mod 156

A:

Q: Use this theorem to solve the following congruences: (1) x° = 2 mod 35 (2) x' = 13 mod 35

A: Since you have asked multiple questions, we can solve first question for you. If you want other…

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: 7. Let p and q be primes and suppose that p = 2q + 1. Let h E F and assume that h 1. Pre that ² h…

A:

Q: he solution of the linear congruence 103x = 444 (mod 999) is ) 111 ) 222 2 333 ) 444

A: We have to find x such that  103x = 444(mod 999)

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: Determine whether the following assertions are true or false. If true, prove and if false, explain…

A: Solution:

Q: Using Fermat's little theorem, find the solutions of the following linear congruences. a) 7x ≡ 12…

A: Fermat's theorem :  Let p be prime number such that gcd(a,p)=1 then ap-1≡1(mod p) Using Fermat's…

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

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Q: (a) 109x = 1 (mod 255) x = (b) 57x = 108 (mod 255) x =

A: X=124

Q: Find the nonnegative integer a less than 28 represented by each of these pairs, where each pair…

A: According to the given information, it is required to find the non-negative integer a less than 28…

Q: 1. The product of (5254)^143 and (14344)^143 under modulo 143 is congruent to ___. a. 98 b. 121 c.…

A: To find: 1) The product of 5254143 and 14344143 under modulo 143 is congruent to ________.…

Q: 9. Determine which of the following congruences have solutions, and find any solutions (between 0…

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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: Solve each of the following congruences. Make sure that the number you enter is in the range [0,M−1]…

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Q: 4. Solve the following: 34x 60 (mod98) 5. Detemine all the possible positive m values greater than 1…

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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: 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: Determine a general congruence condition satisfied by all primes p with p = 3(mod4) such that the…

A: As per the question we are given a number theory problem. Where we have to find a congruence…

Q: e ab then th

A: Given: ak^2+bz^2 =c mod p To prove the given.

Q: 4. Derive each of the following congruences: (b) a² = a (mod 42) for all a.

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

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Q: 2. Find all solutions of each of the following linear congruences. c) 17x = 14 (mod 21) a) 3x = 2…

A: Given: To find the linear congruence of d)15x≡ 9(mod 25)e)128x≡833(mod 1001)f)987x≡610(mod 1597)