1) Use the Euclidean algorithm (write in pseudocode) to find the greatest common divisor of 8 898 and 14 321.
2) Program the Euclidean algorithm in 1) by using C++ programming language.
3) What is the greatest common divisor of 8 898 and 14 321?
4) Next, extend the Euclidean algorithm (write in pseudocode) to express gcd(8 898; 14 321) as a linear combination of 8 898 and 14 321.
5) Continue the programming in 2) to program the Extended Euclidean algorithm in 4).
6) What are the values of the linear combination of 8 898 and 14 321?
Euclidean Algorithm
Euclidean Algorithm is used to find the Greatest Common Divisor (GCD) of 2 numbers. Greatest Common Divisor is the largest digit or number that divides both the number without leaving any remainder (Perfect division).
Rules
Let a, b be two numbers.
1. If either one of a or b is 0 then GCD(a,b)=The number which is zero
For example: if a=0,b=3 then GCD(0,3)=3 because greatest common divisor is 3 only. Similarly GCD(3,0)=3
2. To find GCD(a,b)
Example
Extended Euclidean Algorithm
Extended Euclidean algorithm is used to compute x and y if
ax+by=gcd(a,b)
a and b will be provided to us. We have to reverse the steps that we have done in the euclidean algorithm. That is here we will start from GCD and we do all the steps in the Euclidean algorithm backward.
eg: GCD = 1
321 = 14*22 + 13
14 = 13*1 + 1
13 = 1*13 + 0
1 = 23*14 + -1*321
1. Use the Euclidean algorithm (write in pseudocode) to find the greatest common divisor of 8 898 and 14 321.
Pseudocode Of Program
a=8,b=898
if (a<b) then
swap(a,b)\
while b!=0
temp=a%b
a=b
b=r
end while
print a
a=14,b=321
if (a<b) then
swap(a,b)\
while b!=0
temp=a%b
a=b
b=r
end while
print a
Pseudocode Of Problem
8%898=8
898%8=2
8%2=0
GCD =2
14 % 321 = 14
321 % 14 = 13
14 % 13 = 1
13 % 1 = 0
GCD = 1
2) Program the Euclidean algorithm in 1) by using C++ programming language.
#include <iostream> //Header file for input output using namespace std; int gcd(int a, int b) { //function definition if (b == 0) //if b is zero than return a return a; return gcd(b, a % b); //call the function recursively } int main() { int a , b; // declare 2 variable a & b cout<<"Enter the values of a and b: "<<endl; //print to the user to enter a & b cin>>a>>b; //get value from the user cout<<"GCD of "<< a <<" and "<< b <<" is "<< gcd(a, b); //print gcd value by calling function //gcd(a,b) a and b are the values passed to it return 0; }
OUTPUT
3) What is the greatest common divisor of 8 ,898 and 14, 321?
8%898=8
898%8=2
8%2=0
GCD =2
14 % 321 = 14
321 % 14 = 13
14 % 13 = 1
13 % 1 = 0
GCD = 1
4) Next, extend the Euclidean algorithm (write in pseudocode) to express gcd(8 898; 14 321) as a linear combination of 8 898 and 14 321.
Pseudocode Euclidean algorithm (8 898)
ExtendedEuclidean (a,b) { //definiton of function that accepts 2 parameters a & b
// outputs a triple (p,q,r) where p = gcd(first num,second num) and // p == first num*q + second num*r
if (b == 0) return (a,1,0) ;
(p_1, q_1, r1) = ExtendedEuclidean (8,8%818) ; // a and b will be passed back from this function
p = p_1 ;
q = r_1 ;
r = q_1 - (a div b) * r_1 ; // note: div = integer division return (p,q,r) ;
}
Pseudocode Euclidean algorithm (14 321)
ExtendedEuclidean (a,b) { //definiton of function that accepts 2 parameters a & b
// outputs a triple (p,q,r) where p = gcd(first num,second num) and // p == first num*q + second num*r
if (b == 0) return (a,1,0) ;
(p_1, q_1, r1) = ExtendedEuclidean (14,14%321) ; // a and b will be passed back from this function
p = p_1 ;
q = r_1 ;
r = q_1 - (a div b) * r_1 ; // note: div = integer division return (p,q,r) ;
}
Pseudocode Of Problem
GCD=2
898 = 8*112 + 2
8 = 2*4 + 0
Linear combination : 2 = -112*8 + 1*898
Pseudocode Of Problem
GCD = 1
321 = 14*22 + 13
14 = 13*1 + 1
13 = 1*13 + 0
1 = 23*14 + -1*321
5) Continue the programming in 2) to program the Extended Euclidean algorithm in 4).
#include <bits/stdc++.h> // header file
using namespace std;
int main() { //main function
int x, y;
int a , b ;
cout<<"Enter the values of a and b: "<<endl; //print to
the user to enter a & b
cin>>a>>b; //get value from the user
cout<<"gcd "<<Extended_gcd(a, b, &x, &y);
//print the output from calling function
return 0;
}
int Extended_gcd(int a, int b, int *x, int *y) { //function
definition
if (a == 0) { //checking if a is zero
*x = 0;
*y = 1;
return b;
}
int x1, y1;
int gcd = Extended_gcd(b%a, a, &x1, &y1); //recursive
call
*x = y1 - (b/a) * x1;
*y = x1;
return gcd; //return the calculated gcd back
}
OUTPUT
6) What are the values of the linear combination of 8 898 and 14 321?
Problem
GCD=2
898 = 8*112 + 2
8 = 2*4 + 0
Linear combination : 2 = -112*8 + 1*898
Problem
GCD = 1
321 = 14*22 + 13
14 = 13*1 + 1
13 = 1*13 + 0
1 = 23*14 + -1*321
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