(a) (i) Use Euclid’s algorithm to find a multiplicative inverse of 16 modulo 51, and hence solve the linear congruence 16x ≡ 24 (mod 51).
Added by Emilio M.
Step 1
We have: 51 = 3 * 16 + 3 16 = 5 * 3 + 1 3 = 3 * 1 + 0 So, the greatest common divisor of 16 and 51 is 1. Show more…
Show all steps
Close
Your feedback will help us improve your experience
Madhur L and 52 other Calculus 3 educators are ready to help you.
Ask a new question
Labs
Want to see this concept in action?
Explore this concept interactively to see how it behaves as you change inputs.
Key Concepts
Recommended Videos
solve the linear congruence 16x ≡ 4 mod 29 using the Euclidean algorithm
Madhur L.
(a) (i) Use Euclid's algorithm to find a multiplicative inverse of 19 modulo 45, and hence solve the linear congruence 19x ≡ 24 (mod 45). (ii) Explain why the following linear congruence has no solutions: 5x ≡ 16 (mod 45). (iii) Solve the linear congruence 10x ≡ 15 (mod 45). (b) The affine cipher E is given by E(x) ≡ 15x - 7 (mod 26). The conversion table for letters and numbers is shown below. A B C D E F G H I J K L M 0 1 2 3 4 5 6 7 8 9 10 11 12 N O P Q R S T U V W X Y Z 13 14 15 16 17 18 19 20 21 22 23 24 25 (i) Show that 7 is a multiplicative inverse of 15 modulo 26. (ii) Suppose that you receive the enciphered message 14, 7, 6. Decipher the message and deduce the word that was sent.
Supreeta N.
(i) Use the Euclidean Algorithm to compute gcd(2,11), and then express gcd(2,11) as a linear combination of 2 and 11. (ii) Hence, find the inverse of 2 modulo 11 (iii) Hence, solve the linear congruence 2x ≡ 9 mod 11
Suman K.
Recommended Textbooks
Calculus: Early Transcendentals
Thomas Calculus
Transcript
18,000,000+
Students on Numerade
Trusted by students at 8,000+ universities
Watch the video solution with this free unlock.
EMAIL
PASSWORD