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Largest Coprime Set Between two integers

  • Last Updated : 08 Dec, 2021

Given two integers L and R which denotes a range, the task is to find the largest co-prime set of integers in range L to R. 
Examples: 
 

Input: L = 10, R = 25 
Output: 10 11 13 17 19 21 23
Input: L = 45, R = 57 
Output: 45 46 47 49 53 
 

 

Approach: The idea is to iterate from L to R and try to place the integer in a set such that the Greatest common divisor of the set remains 1. This can be done by storing the LCM of the set and each time before adding the element into the set check that the GCD of the number with LCM of the set remains 1. Finally, find the largest such set of the integers.
For Example: 
 

Let L = 10, R = 14

Element 10:
// Co-prime Sets 
S = {{10}},
LCM of Co-prime sets
A = {10}

Element 11:
// Element 11 can be added to
// the first set
S = {{10, 11}}
A = {110}

Element 12:
S = {{10, 11}, {12}}
A = {110, 12}

Element 13:
S = {{10, 11, 13}, {12}}
A = {1430, 12}

Element 14:
S = {{10, 11, 13}, {12}, {14}}
A = {1430, 12, 14}

 

C++




// C++ implementation to find
// the largest co-prime set in a
// given range
#include <bits/stdc++.h>
using namespace std;
 
// Function to find the largest
// co-prime set of the integers
void findCoPrime(int n, int m)
{
    // Initialize sets
    // with starting integers
    vector<int> a = { n };
    vector<vector<int> > b = { a };
 
    // Iterate over all the possible
    // values of the integers
    for (int i = n + 1; i <= m; i++) {
 
        // lcm of each set in array
        // 'b' stored in set 'a'
        // so go through set 'a'
        for (int j = 0; j < a.size(); j++) {
            // if there gcd is 1 then
            // element add in that
            // array corresponding to b
            if (__gcd(i, a[j]) == 1) {
 
                // update the new lcm value
                int q = (i * a[j]) / __gcd(i, a[j]);
 
                b[j].push_back(i);
                a[j] = q;
            }
        }
        a.push_back(i);
        b.push_back({ i });
    }
    vector<int> maxi = {};
    for (int i = 0; i < b.size(); i++) {
        if (b[i].size() > maxi.size()) {
            maxi = b[i];
        }
    }
    for (auto i = maxi.begin(); i != maxi.end(); i++) {
        cout << *i << " ";
    }
    cout << endl;
}
 
// Driver Code
int main()
{
    int n = 10;
    int m = 14;
    findCoPrime(n, m);
    return 0;
}
 
// This code is contributed by rj13to.

Python




# Python implementation to find
# the largest co-prime set in a
# given range
 
import math
 
# Function to find the largest
# co-prime set of the integers
def findCoPrime(n, m):
    # Initialize sets
    # with starting integers
    a =[n]
    b =[[n]]
     
    # Iterate over all the possible
    # values of the integers
    for i in range(n + 1, m + 1):
     
        # lcm of each list in array
        # 'b' stored in list 'a'
        # so go through list 'a'
        for j in range(len(a)):
             
            # if there gcd is 1 then
            # element add in that
            # list corresponding to b
            if math.gcd(i, a[j])== 1:
                 
                # update the new lcm value
                q =(i * a[j])//math.gcd(i, a[j])
                r = b[j]
                r.append(i)
                b[j]= r
                a[j]= q
        else:
            a.append(i)
            b.append([i])
    maxi = []
    for i in b:
        if len(i) > len(maxi):
            maxi = i
    print(*maxi)
     
# Driver Code
if __name__ == "__main__":
    n = 10
    m = 14
    findCoPrime(n, m)
Output: 
10 11 13

 


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