Q1Exercise 1: Modular arithmetic Explain how to find the smallest integerm> 0 such that the congruence class of m coincides with thecongruence class of...1. ... 123456789 modulo 2.2. ... 6333 modulo 9.3.4.5.... [84]7 [9023]7....3111+123456789 modulo 3.[14]4[19]4.

Given: 123456789 modulo 2To find: The smallest integer m≥0 such that the congruence class of m…

Answered: Exercise 1: Modular arithmetic Explain…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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2. Which of the following option is suitable, if A is “10110110”, B is “11100000” and C is “10100000” A. C=A or B B. C = ~A C. C = ~B D. C=A and B
(b) Modular arithmetic Let x, y, N € N be n-bit integers such that N > 2. • If x < N, the value x" mod N can be computed using the following recursive formula: x" mod N = 1 X gcd (x, y) = The number of bit operations when using this method is O(.....). • If x ≥ y, the value gcd(x, y) can be computed using the following recursive formula: , if y = 0 , if y ≥ 1 -₁ , if y = 0 , if y = 1 , if y ≥ 2 even , if y ≥ 2 odd The number of bit operations when using this method is O(.....).
Define the fibonacci numbers by f(0) = 0; f(1) = 1; and then recursively for natural numbers n ≥ 2 by f(n) = f(n-1) + f(n-2). a) Write out the first 7 fibonacci numbers. I b) Make a conjecture about which numbers n will give us that f(n) will be an even number. c) Prove an exact characterization of these numbers using strong induction. (i.e. show that if your characterization is true up to some point, then it would be true just after that)
5. (a) Use modular arithmetic to show that if an integer a is not divisible by 3, then a2 = 1 (mod 3) (b) Use this result to prove that in any Pythagorean triple (x, y, z), either x or y (or both) must be divisible by 3
Partition (Z/19Z)* into equivalence classes determined by the identi- fication of elements of the same order. What are the primitive roots modulo 19?
discrete math
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Exercise 1: Modular arithmetic Explain how to find the smallest integer m>0 such that the congruence class of m coincides with the congruence class of... 1. ... 123456789 modulo 2.