The Euclidean Algorithm is a way of finding the greatest common divisor d of two natural numbers n and m. Trace through the Euclidean Algorithm 5.14 for the following pairs of numbers:(a) m=9,n=12(b) m=42,n=24 **Algorithm 5.14 The Euclidean Algorithm****Preconditions:**\( m, n \in \{0, 1, 2, 3, \ldots \} \)**Postconditions:**\( d \) is the greatest integer such that \( d \mid m \) and \( d \mid n \).**Euclidean Algorithm Steps:**1. **Initialize:** - Set \( d \leftarrow m \) - Set \( e \leftarrow n \)2. **Loop until \( e \neq 0 \):** - Compute \( r \leftarrow d \mod e \) (remainder of \( d \) divided by \( e \)) - Update \( d \leftarrow e \) - Update \( e \leftarrow r \)This algorithm effectively finds the greatest common divisor (GCD) of the given integers \( m \) and \( n \). The algorithm repeatedly replaces the larger number by its remainder when divided by the smaller number until the smaller number reaches zero. The last non-zero remainder is the GCD of the two original numbers.

Let's go through the Euclidean Algorithm for the given pairs of numbers:(a) m = 9, n = 12:Step 1:…

The Euclidean Algorithm is a way of finding the greatest common divisor d of two natural numbers n and m. Trace through the Euclidean Algorithm 5.14 for the following pairs of numbers: (a) m=9,n=12 (b) m=42,n=24

The Euclidean Algorithm is a way of finding the greatest common divisor d of two natural numbers n and m. Trace through the Euclidean Algorithm 5.14 for the following pairs of numbers: (a) m=9,n=12 (b) m=42,n=24

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Related questions

Q: Use Euclid's algorithm to find the greatest common divisor of 45 and 72. Hence find the lowest…

Q: Use the extended Euclidean algorithm to find the greatest common divisor of 8,820 and 1,005 and…

A: To use the Extended Euclidean Algorithm to find the gcd8820, 1005 and express the linear combination…

Q: The Euclidean Algorithm is used to find the greatest common divisor of 7320 and 873. What is the…

Q: (a) Use the division algorithm to show that any number N can be written as either N = 3k, N = 3k +…

A: (a) Use the division algorithm to show that any number N can be written as either N = 3k, N = 3k +…

Q: Find the gcd of 21-38i and 113-59¿ in two ways: first, by factorizing both numbers into Gaussian…

A: Since you have asked multiple questions, we will solve the first one for you. If you want any…

Q: (i) Use the Euclidean Algorithm to find gcd(4635, 873). (ii) Find integers m and n such that 4635m +…

Q: Use tox Euclidean algorithm to computeo the greatest common divisor of 1197 and 14280, and to…

Q: Introductory to Number Theory: The Euclidean Algorithm Give an example of positive integers p, a,…

Q: 1) Use the Euclidean algorithm to compute the greatest common divisor of the integers a = 469 and b…

Q: Show that for every positive

Q: (a) Find an inverse for 73 modulo 66o. First use the extended Euclidean algorithm to find the…

A: Given in box all are green tick are correct .

Q: Use the extended Euclidean algorithm to find the greatest common divisor of the given numbers and…

A: Given: Linear combination 5055s+1905t To find: The gcd and the value of variables s and t

Q: Compute that greatest common divisor of 142 and 344 using the Euclidean algorithm. What is the…

Q: 3. For the numbers 360 and 70, use the Euclidean Algorithm (augmented or otherwise) to find the…

Q: a) Let D be the greatest common divisor of 345 and 270. Use Euclidean Algorithm to compute the value…

Q: 4. Find the gcd of 182 and 1001 by Euclidean Algorithm, and find integers m, n such that the gcd…

A: I have given the answer in the next step. Hope you understand that

Q: Construct a trace table to trace the action of the Euclidean Algorithm for the imput variabless 1570…

A: To Construct: A Trace table for the action of Euclidean algorithm for inputs a=1570, b=488.

Q: 3. Find the gcd of 29 and 115 by Euclidean Algorithm, and find integers m, n such that the gcd…

Q: USE PENCIL AND PAPER, DONT USE AI .Compute the greatest common divisor gcd of 3 and 81 using the…

A: Euclidean Algorithm:Steps to Solve gcd(3, 81) Using the Euclidean Algorithm:

Q: Find the smallest non negative value of D such that ED is congruent to 1 (mod b) when b=616…

A: As given in the hint the idea is to use the extended Euclidean algorthim to solve the congruence.…

Q: i just need help in the step 6 and conclusion. im not sure how to start it.

Question

The Euclidean Algorithm is a way of finding the greatest common divisor d of two natural numbers n and m. Trace through the Euclidean Algorithm 5.14 for the following pairs of numbers:

(a) m=9,n=12

(b) m=42,n=24

**Algorithm 5.14 The Euclidean Algorithm**

**Preconditions:**
\( m, n \in \{0, 1, 2, 3, \ldots \} \)

**Postconditions:**
\( d \) is the greatest integer such that \( d \mid m \) and \( d \mid n \).

**Euclidean Algorithm Steps:**

1. **Initialize:**
- Set \( d \leftarrow m \)
- Set \( e \leftarrow n \)

2. **Loop until \( e \neq 0 \):**
- Compute \( r \leftarrow d \mod e \) (remainder of \( d \) divided by \( e \))
- Update \( d \leftarrow e \)
- Update \( e \leftarrow r \)

This algorithm effectively finds the greatest common divisor (GCD) of the given integers \( m \) and \( n \). The algorithm repeatedly replaces the larger number by its remainder when divided by the smaller number until the smaller number reaches zero. The last non-zero remainder is the GCD of the two original numbers.

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

SEE SOLUTION Check out a sample Q&A here

Step 1: Discription for the step 1

VIEW

Step 2: Discription for the step2

VIEW

Solution

VIEW

Step by step

Solved in 3 steps

SEE SOLUTION Check out a sample Q&A here

Similar questions

13. a. Find the GCD of the following pairs of numbers using the Euclidean algorithm. (3360,4576) b. Find the LCM of the following pairs of the numbers. (144, 180)
Use Euclid's algorithm to find gcd(340, 144). Hence find Icm (340, 144).
For each pair of integers a and b, use the Euclidean Algorithm to find the GCD(a, b). Then use Theorem 92 to find the LCM(a, b) a. a = 48 and b = 54 b. a 330 and b = 156 c. a = 1188 and b = 385
If you are using the Sieve of Eratosthenes to find all the prime numbers between 50 and 150, how many times are you going to perform the algorithm before you can con- clude that all the numbers which remains un-striken are prime numbers. 12 2 O 11
Find the remainders at each of the four steps of the Euclidean algorithm in the process to find the GCD of 1053 and 80. What is the GCD Of 1053 and 80?