(a) (i) Use Euclid's algorithm to find a multiplicative inverse of23 modulo 75, and hence solve the linear congruence23x = 19 (mod 75).(ii) Explain why the following linear congruence has no solutions:25x = 30 (mod 75).(iii) Solve the linear congruence45x = 60 (mod 75).

Answered: Image /qna-images/answer/8066f733-4e7b-4fc6-a889-a6e30ddabc60.jpg

Answered: (a) (i) Use Euclid's algorithm to find…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter2: The Integers

Section2.7: Introduction To Coding Theory (optional)

Problem 12E: Suppose that the check digit is computed as described in Example . Prove that transposition errors...

See similar textbooks

Similar questions

Suppose that the check digit is computed as described in Example . Prove that transposition errors of adjacent digits will not be detected unless one of the digits is the check digit. Example Using Check Digits Many companies use check digits for security purposes or for error detection. For example, an the digit may be appended to a -bit identification number to obtain the -digit invoice number of the form where the th bit, , is the check digit, computed as . If congruence modulo is used, then the check digit for an identification number . Thus the complete correct invoice number would appear as . If the invoice number were used instead and checked, an error would be detected, since .
Use your work from problem 2 and the method used in the example at the end of Lecture 16 to solve the congruence: 7x5 is congruent to 6 (mod 17) .
a) Rewrite 17x=3 (mod 53) as linear combination. b) Using Euclidean Algorithm to find the gcd(17, 53). c) Using recursive Euclidean Algorithm to express the gcd(17, 53) as a linear combination of 17 and 53. d) Using the Linear Combination you found in part c to find the multiplicative inverse of 17(mod 53). e) Using the inverse you find in part d to solve 17x=3 (mod 53) , make sure you reduce your answer to LNR form. You can use " = " as "".
Find the solutions of the congruence x2 ≡ 16 (mod 105).
How do I use the Chinese Remainder Thm to solve this system of congrunces below?x ≡ 1 (mod 3)x ≡ 4 (mod 7)x ≡ 8 (mod 11)
For the congruence 17x ≡ 9 (mod 276), write down an equivalent system of congruences. Hence solve the congruence 17x ≡ 9 (mod 276).
Find all values of n with 0 ≤ n ≤ 35 such that the congruence 24x ≡ n (mod 36) has a solution. Showall the necessary work to obtain your answers.
Modular division can be performed by considering the related multiplication problem. For instance, if 5 ÷ 7 = x, then x · 7 = 5. Similarly, the quotient (5 ÷ 7) mod 8 is the solution to the congruence equation x · 7 ≡ 5 mod 8, which is 3. Find the given quotient. (8 ÷ 5) mod 9
Solve the congruence x^2 ≡ 1 (mod 105)
Solve x2+3x+2 is congruent to 0 (mod 55)
2. Solve the linear congruence 35x ≡ 77(mod89).
Solve for d, 19d ≡ 1 mod (37422000) using the "Extended Euclid's Algorithm"

Recommended textbooks for you

Elements Of Modern Algebra

Algebra

ISBN:

9781285463230

Author:

Gilbert, Linda, Jimmie

Publisher:

Cengage Learning,

Elements Of Modern Algebra

Algebra

ISBN:

9781285463230

Author:

Gilbert, Linda, Jimmie

Publisher:

Cengage Learning,

SEE MORE TEXTBOOKS

(a) (i) Use Euclid's algorithm to find a multiplicative inverse of 23 modulo 75, and hence solve the linear congruence 23x = 19 (mod 75). (ii) Explain why the following linear congruence has no solutions: 25x = 30 (mod 75). (iii) Solve the linear congruence 45x = 60 (mod 75).