
Please solve the above 4 questions.
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Please solve the above 4 questions. 1. Using the extended Euclidean Algorithm, find all solutions of the linear congrue...
Question 1. (a) Find the greatest common divisor of 10098 and 3597 using the Euclidean Algorithm. (b) Find integers a and a2 with 1009801 +3597a2 = gcd(10098,3597). (c) Are there integers bı and b2 with 10098b1 + 3597b2 = 71? Justify your answer. (d) Are there integers ci and c2 with 10098c1 + 3597c2 = 99? Justify your answer. Question 2. Consider the following congruence. C: 21.- 34 = 15 (mod 521) (a) Find all solutions x € Z to...
Please write neatly and clearly, show all work. Thank you! (I've
been stumped for awhile)
(1 point) Find the smallest positive integer x that solves the congruence: 11x = 4 (mod 68) x = (Hint: From running the Euclidean algorithm forwards and backwards we get 1 = s(11) + +(68). Find s and use it to solve the congruence.)
Please show question 1 (all parts). Thank
you!
1. Using the Euclidean algorithm to find the ged of following pairs. Write down the ged as a linear combination of given pairs (a) 524 and 148 in Z (b)33 + 2r +1 and 2 +1 in Zs[] (c) 3 +2r +1 and 1 n Z[] 2. Compute 42001 in Z5 3. Use principal of induction show that 10" 1 mod 9 4. Show that every odd integer is congruent to 1...
1. Solve each linear congruence for all integers x so that 0 sx <m a) 11x 8 (mod 57) b) 14x 3 (mod 231)
Solve the following system of equations and find all congruence class solutions if any exist; if no solution exists, explain why not: x24 (mod 35) 3.x 15 mod 21)
3. (16 points) Solve the system of linear congruences using the Chinese Remainder Theorem. 4 (mod 11) a 11 (mod 12) x=0 (mod 13) b. (6 pts) Find the inverses n (mod 11), n21 (mod 12), and nz1 (mod 13). Using these ingredients find the common solution a (mod N) to the system. c. (4 pts) 4. (8 points) What is 1!+ 23+50! congruent to modulo 14?
Find all solutions to the congruence x2+ x+ 1≡0 mod 91. (Hint:factor the modulus, use trial and error to find the solutions modulo the factors, and the CRT to combine the results into solutions to the original equations.)
In this problem you will implement an algorithm for computing all the square roots of a congruence class in ℤ/nℤ, given a complete factorization of n into its distinct prime factor powers (assuming all the prime factors are in 3 + 4ℤ). a) Implement a Python function sqrtsPrime(a, p) that takes two arguments: an integer a and a prime number p. You may assume that a and p are coprime. If p is not in 3 + 4ℤ or a...
(**) In #5-#6, find all solutions of under determined linear system by using the row operations: (2. 0:01 (0.21 -4.02 +2.13 +24 +02 -13 +2.04 +0.02 +0x3 +344 = 11 = 5 = 9 2 0.0 ( 0.1 +2.02 +02 0.02 -13 -4.03 0.13 +0.04 +3.25 +0.04 +45 + +2:05 = 7 = -1 = 1
(1) Define what an integral domain is. (2) Find all solutions to r? + 5x + 6 = 0 in Za. (3) Find all units in Z14 (4) Solve the equation 3.x = 2 in Zs. (5) Find the remainder of 512 when it is divided by 11. IR Camonte (12) Using this, determine 5 (mod 12).