Show that if p is a prime number then every equation of the form ax ≡ b mod p with a 6≡ 0 mod p has at most one solution. Give examples to show that linear equations mod n (for n not prime) that can have zero or multiple solutions.

Q: Prove that x113+x-1≡0(mod 113) has no solutions in x.

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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: Q16

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

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Q: + k-a a 거+a a-k

A: Given:   ya=-xk-a+1 To Prove: y=aa-kx+a

Q: 7) Note that -2 is a square modulo 11. Indeed, 82 +2 = 6.11. Use this information, and the method of…

A: To use method of descent to construct a solution of x2+2y2=11 Now, it is known that -2 is a square…

Q: 2. Let a be a positive integer. Explain why the linear congruence ax = b (mod a + 1) has a solution…

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Q: Q2 Consider nonnegative integer solutions of the equation…

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

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Q: 36. Find all solutions to each of the following linear equations: a. 3x= 4 (mod 5) b. 4x = 4 (mod 6)…

A: a)  In modular arithmetic, you can find the solution by multiplying both sides of the congruence by…

Q: How many nonnegative integer solutions are there to the equation Xị + x2 + x3 + x4 + x5 = 40 if we…

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Q: Solve the system 3x + 2y = 5 2x + 7y= 9 mod 17 mod 17

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Q: Let Describe all solutions of Ax = 0. x = x2 0000 +x3 A = 1 2 +x4 -4 8 10 8 -20 - 16 0000

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Q: а). Solve the linear congruence of 231x = 147 mod 598 .

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Q: Find a general solution of ди ди 2- + 3 2u ar

A: Given the pde

Q: 2yy" = 1 + (y')² 2

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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: 1. Find all r €Z satisfying each of the following equations. (a) 3r = 2 (mod 7) (d) 9r = 3 (mod 5)…

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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: (d) 9r = 3 (mod 5) (e) 5x = 1 (mod 6) (f) 3x = 1 (mod 6)

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Q: 2x = 4 (mod 8) a. has no solution b. has infinite solutions c. has a unique solution modulo 8 which…

A: Here 2x≡4 (mod 8). Hence 2x=4+8k, for some integer k. Dividing by 2, x=2+4k, for some integer k.…

Q: Find all integer solutions r, y,p to the equation I+y+x² + y? = p such that p is a prime number.

A: In this equation, two integers and their squares are added on the LHS. Squares of integers are…

Q: Solve the congruence3x−7 = 2 (mod 16). Your solution should be positive and reduced modulo16.

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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: Prove that if a > 0, and b is an arbitrary real number then x³ + ax + b= 0 has exactly ne real…

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Q: For which positive numbers a is it true that ax >= 1+x for all x?

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Q: 5. Solve the given ODE using the method of undetermined coefficients. y" − 4y = x² + 4x − 3 -

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

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

Q: Let Describe all solutions of Ax = x = x₂ +x3 +x4 1 -₁2 50]. 4 -5 -10 8 10 A =

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Q: Let AABC and ADEF are similar triangles and their areas be respectively 64 cm and 121 cm. cm? If EF…

A: as per bartleby guidelines i can do first question only

Q: Solve the systems of linear congruence equations.

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Q: 14. Give a geometric explanation of why 2π/n 1-2 0 r² do = лr² 2 for any positive real number r and…

A: n2∫02πnr2dθ=πr2 For any positive real number r and any positive integer n. Prove this geometrically.…

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

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

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Q: Show that th equation x" + y² = z" (mod 5) has nonzero solutions in Zz if and only if n is odd.

A: Given: xn+y2n≡zn(mod 5)

Q: Let, a1= 3 and for n ≥ 2, an = 2an-1 + 1, express an in terms of n.

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Q: Which choice gives an example that supports the conjecture, and a counterexample that shows the…

A: The objective is to select the correct option from the given options. Given that, for any real…

Q: 9] Using the discriminant, choose the pair(s) of integers that can be substituted into the equation…

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Q: Find the number of 5-tuples (a, b, c, d, e) of positive odd integers that are solutions to the…

A: The idea is to transform this into a known counting problem

Q: or what values of k, x=4 will be a solution for 3x²+4x = k mod31?

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Q: Why does the linear congruence 25x ≡ 30 (mod 75) have no solutions?

A: To show that the linear congruence 25x≡30mod 75 have no solutions.