The linear congruence in two variables ax + by ≡ c (mod m), where a, b, c, and m are integers, m > 0 with d = (a, b, m), has exactly dm incongruent solutions if d|c, and no solutions otherwise. Find all solutions of the following linear congruences in two variables: (i) 2x + 3y ≡ 1 (mod 7) (ii) 10x + 5y ≡ 9 (mod 15)

Q: The equation 20x=30 (mod 70) has: 10 non-congruent solutions modulo 70 None of these no solution…

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Q: Find all solutions to the system of congruences: x ≡ 7 mod 12 x ≡ 37 mod 45 2x ≡ 134 mod 150

A: The given problem is to find the all solutions which satisfies the given system of linear…

Q: 2r = 3 4x = 2 3x 2 (mod 5) (mod 6) (mod 7).

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Q: Q2 Consider nonnegative integer solutions of the equation…

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Q: How many non-negative integer solutions (O is allowed) are there to r +y+ z = 24?

A: The given equation is, x+y+z= 24 We know that, For equation,  x1+x2+......xr=n  There are n+r-1Cr-1…

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

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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: he linear congruence 132x ≡ 76 (mod 100) has a solution. Select one: True False

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

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Q: Find the common solutions to the congruences 2x 3x 4x 1 (mod 5) 9 (mod 6) 1 (mod 7).

A: Given : 2x≡1 (mod 5)3x≡9 (mod 6)4x≡1 (mod 7) To find : The common solutions to the congruences. For…

Q: Solve the system of congruences 2x ≡ 1 (mod 5) 3x ≡ 9 (mod 6) 4x ≡ 1 (mod 7) 5x ≡ 9 (mod 11) Hint:…

A: We can solve this system of congruences using the Chinese Remainder Theorem. First, let's write each…

Q: Solve the congruence 11x = 45 (mod 97) where x is an integer in the interval [0,97).

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Q: Solve the givnen congruence x2 ≡ 9 (mod 600).

A: The given problem is to solve the given congruence equation. x2=9 (mod 600).

Q: 3. Solve the system of congruences x = 1(mod 5) x 2(mod 6) r= 3(mod 7)

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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: 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: 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: 11. Solve the systems of congruences: a. x = 7(mod 11), x = 4(mod 24) b. 3x = 11(mod 17), 2x = 3(mod…

A: The given problem is to solve the given system of congruence equations. Given system of congruence…

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: 1. Solve each of the linear congruence equations or verify that it has no solution. (i). 3r = 4(mod…

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Q: Find all the non-congruent solutions to  4x ≡ 6(mod 21)  express solutions using the stand…

A: Introduction: A linear congruence is generalized as ax≡bmod n. The greatest common divisor of a and…

Q: Solve the systems of linear congruence equations.

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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: 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: Explain why the following linear congruence has no solutions:   5x ≡ 16 (mod 45).

A: Explain why the following linear congruence has no solutions: 5x ≡ 16 (mod 45).

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

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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: 2X−4Y+6Z≡3 (mod8)  2X−4Y+6Z≡3 (mod8) 3X+6Y−9Z≡1 (mod19) 3X+6Y−9Z≡1 (mod19)

A: This is a problem of modular arithmetic, number theory.

Q: 96. Solve the congruence 13 – 29x2 + 35x + 38 = 0 (mod 195).

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

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Q: Solve the linear congruence 7а + 3у %3D 10 (тod16).

A: Consider the linear congruence 7x + 3y ≡ 10 mod 16.Since gcd (7, 3) = 1 we know at least one…

Q: he solutions of the equation 7? ≡ 28 (mod 119).

A: We have to find out all the solutions of the equation 7?≡28(mod 119) ax≡b(mod n) has solution iff…

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.

Q: Solve the simultaneous system of congruences: 3x = 7 mod 11 5x 10 mod 15

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Q: where every entry on the diagonal of the matrix in the last line is 11, and every entry below the…

A: We have to find the values of a,b,c, and d.

Q: Solve 55x ≡ 2(mod243) by inverse method.

A: Given 55x≡2mod 243. Solve this congruence using the inevrse method.

Q: 2. Solve the simultaneous congruences x = 4 (mod 5) and 2x = 3 (mod 7)

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Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.