Determine whether the given congruence(s) is/are solvable and find all incongruent solutions if they exist. (a) 75x = 77 (mod 111) (b) 28x = 48 (mod 68) (c) x57x8 + 15x + 1 = 0 (mod 16) (Hint: Consider the cases (i) 2x and (ii) 2 | x.)

Q: What is the number of solutions to the congruence in Z125? x³ + x² + 3 = 0 (mod 125)

A: As per the question we have to solve the following polynomial congruence equation : x3 + x2 + 3 ≡ 0…

Q: (10) Find all solutions of the congruence 2 = 22 (mod 53). C

A: We shall be using primitive roots to solve this congruence.

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

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Q: Show that f[x0, x1, x2] = f[x2, x1, xo].

A:

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

A:

Q: 4. Solve each of these congruences using the modular inverses found in the previous problem. (a) 7x…

A: The given congruence equations are (a)    7x=2 (mod 26) (b)   19x=4  (mod 141)

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

A:

Q: Find the smallest positive integer x that solves the congruence: 4x7 (mod 23)

A: we solve the modular equation

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

A:

Q: Find all solutions of each congruence and explain why these are all the solutions. (It is possible…

A:

Q: (73 – 98)(mod 3) = (mod 3)

A: GIiven, a expression using modulo function.

Q: Find all solutions in {0, 1,2, ..., 12} to x26 + 15x13 = 12 mod 13.

A:

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: Can you please show me step by step for this math that makes me undertandable. Q: Determine which…

A: (a) For the congruence 4x ≡ 5 (mod 6), we need to find a solution for x such that when 4x is…

Q: x43 = 47 (mod 713)

A: Step:-1 By Chinese remainder Theorem (CRT),  x43 ≡ 47 mod 713 if and only if x43 ≡47 mod 23 ⇒x43 ≡1…

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).

A:

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: B- State Fermat little theorem, then find √√27 mod 31. Find (52525).

A:

Q: 3. Solve the following congruence equations. (a) 20 984 (mod 1984). 11r + 5y = 7 (mod 20) 6x + 3y =…

A:

Q: 2. Euclidean Algorithm Use the Euclidean algorithm to find gcd(12345, 678).

A: We need to find the GCD(12345,678)=?

Q: Does x2 +3x+7=0 mod 31 have solutions? IF

A: This follows in the same way as we study quadratic equations. Here we use facts about quadratic…

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: (3) If today is Thursday what day of the week will it be 31 days from today? (4) If it is currently…

A:

Q: 2. Encrypt the message STOP POLLUTION by translat- ing the letters into numbers, applying the given…

A: let's encrypt the message "STOP POLLUTION" using the given encryption functions one by one.For the…

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: 2a + 56+c —а — 36 — 2с 3a + 76 +c -4a – 96 –c 2. T(а + bx + сх*). |

A: The objective is to write down the matrix of the given linear transformation withrespect to the…

Q: Hellcopter Trip Whale Watching Adventure Volcano Tour Hours Cost ($) Hours Cost ($) Hours Cost ($)…

A: We can use linear equations to solve these types of problems. There are different methods available…

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: Solve the following linear congruences (you may if you wish, find some solutions by inspection): (a)…

A:

Q: . Find all the solutions of 3x² + 2x + 1 8 (mod 63).

A: Given a quadratic congruence in the form:Ax2+Bx+C=0(mod n)multiply the above equation by…

Q: 6. Find all incongruent solutions to each congruence. Show your steps. (a) 135x = 108 (mod 612) (b)…

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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: Use Hensel Lifting (twice) to find a solution to x² = 7 (mod 374).

A: According to the given information, it is required to use Hensel lifting method to find the…

Q: 3. Solve the following congruences. † (a) 10x = 5 (mod 21) (b) 10x = 5 (mod 15) † (c) 10x = 4 (mod…

A:

Q: y" + 4y' + 13y = te-t %3D y(0) = 0 y'(0) = 2

A: