hinese Remainder Theorem Solve for the smallest value of x in the following linear congruences. Show your solutions. 1. x ≡ 3 mod 6 and x ≡ 4 mod 7 2. x ≡ 3 mod 5 and x ≡ 2 mod 3 3. x ≡ 7 mod 9 and x ≡ 1 mod 5 4. x ≡ 10 mod 17 and x ≡ 1 mod

Q: Find all the solutions of the congruence equation x6-2x5= 35 (mod 6125).

A: Consider the given congruence equation. x6-2x5≡35mod 6125. Now, the prime factorization of 6125 is:…

Q: Find all solutions for x, up to congruence.  x ≡ 2^63 (mod 61).

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

A: We have to solve the above quadratic congurence.

Q: Q2: Find the solution of 143x³ + 22x¹ + 5x² + 3x + 1 = 0 (mod 11), if any.

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Q: from x = 2 to x = 10 and x; is any value of x in Ex¡. Express lim s as a definite integral. n-x n D…

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

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Q: Find all solutions for x, up to congruence. x ≡ 1 mod 4 and x ≡ 7 mod 13.

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Q: Show that the polynomial 1+x+x² +...+xP-1 . where p is a prime number, is irreducible over the field…

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Q: Find all four solutions to x2 =6 (mod 19 x×23) with 1<x< 437.

A: Solution:-

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

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Q: If p times tne pth term of an A.P. is equal to q times the qth term of the same AP., then how that…

A: Given that If p times the pth term of an A. P. is equal to q times the qth term of same A. P.  To…

Q: x^4 + 2x +12 = 0( mod24) All that is not equivalent to polynomial congruence Find solutions.

A: We must identify values of that fulfill the congruence equation x^ 4 + 2x +12 ≡ 0 (mod 24) in order…

Q: 2. ) For each statement, Determine if it is True. If it is true then prove it. If it is False then…

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

Q: If p(q-r) x² + q (r-p)x+r(p-q) = 0 has equal roots, then 2 1 T= 9 prove that Р + 1 r

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Q: Which of the following linear congruence has 2 incongruent solutions? (a) 3x = 9 (mod21) (b) 14x =…

A: to solve linear congurence , ax≡b(mod m) .........(1) the first stem is to reduce the congurence  by…

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

A: Topic:- algebra

Q: Find the solutions of a3 – 24.x2 + 54 = 0( mod 25).

A: want to find solution of x3-24x2+54≡0(mod 25)note that1 a≡b(mod m) f and only if m|a-b(i.e a-b=mk…

Q: Find all solutions for x, up to congruence. (b) x ≡ 11 mod 142 and x ≡ 25 mod 86.

A: Given: (b) x≡11 mod 142 and x≡25 mod 86 To find: All solutions for x, up to convergence.

Q: If p≥2 is a prime and a,b ∈Z(p) with a̸ = 0 ,we say that b/p a is the solution x∈Z(p) of the…

A: This is a question of Number Theory.

Q: 3. Show that all integers n ≥ 2 can be expressed as 2x + 3y for some nonnegative integers r and y.

A: Mathematical induction is a proof technique used in mathematics to establish the truth of a…

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.) Determine if x< = 2 (mod 7817) has solutions. b.) Determine if x< = 3760 (mod 4861) has…

A: x2≡2(mod 7817) has a solution if and only if  2 is a quadratic residue of 7817. Euler's criterion…

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

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

Q: If w and af are the nonreal cube roots of unity and [1/(a + c)] + [1/(b + w)] + [1/(c + w)] = 2w²…

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Q: (a) 64x 56 (mod 52) (b) 14x = 35 (mod 56)

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Q: Show that n := e2™i/n is a primitive nth root of unity in C. Verify that (S3 - )? = -3 in C. Verify…

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Q: 2/5+5/2 2/5-5/2 Express in the form g +rvs where q,r and s are rational numbers.

A: Simply rationalize the given expression and simplifie.

Q: Describe geometrically the effect of the transformation T Let A = -(1) Define a transformation T by…

A: We are authorized to answer one question at a time since you have not mentioned which question you…

Q: If possible, solve the following linear congruences. (a) 42x = 1014 (mod 91) (b) 42x = 1015 (mod 91)

A: If possible, we have to solve the following linear congruences.

Q: Find all the solutions of 15z = 27 (mod 18).

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Q: Find the solutions of the congruence x2 ≡ 16 (mod 105).

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Q: y²=x³-5x+3 (mod5). solve for all x and y, when x is at least 0, and y is at most 4

A: Using congruence and modular arithmetic we solve the given equation. We substitute the value of x…

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: ind all solutions of the linear congruence: 9x = 5 mod (14)

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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: The complete set of the real values of k for which the equation x + kx +2k 0 has real distinct roots…

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Q: Give all non-negative solutions for 10r =-17 mod 4.

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