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Divide N into K unique parts such that gcd of those parts is maximum

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  • Last Updated : 09 Nov, 2021

Given a positive integer N, the task is to divide it into K unique parts such that the sum of these parts is equal to the original number and the gcd of all the parts is maximum. Print the maximum gcd if such a division exists else print -1.
 

Examples: 

Input: N = 6, K = 3 
Output:
Only possible division with unique 
elements are (1, 2, 3) and gcd(1, 2, 3) = 1.
Input: N = 18, K = 3 
Output:

Naive approach: Find all the possible divisions of N and compute the maximum gcd of them. But this approach will take exponential time and space.
Efficient approach: Let the divisions of N be A1, A2……..AK 
Now, it is known that gcd(a, b) = gcd(a, b, a + b) and hence, gcd(A1, A2….AK) = gcd(A1, A2….AK, A1 + A2…. + AK) 
But A1 + A2…. + AK = N and hence, the gcd of the divisions will be one of the factors of N
Let a be the factor of N which can be the possible answer: 
Since a is the gcd, all the division will be the multiples of a and hence, for K unique Bi
a * B1 + a * B2……. + a * BK = N 
a * (B1 + B2…….+ BK) = N 
Since all the Bi are unique, 
B1 + B2…….+ BK ≥ 1 + 2 + 3 ….. + K 
B1 + B2…….+ BK ≥ K * (K + 1) / 2 
Hence, all the factors of N whose complementary factor is greater than or equal to K * (K + 1) / 2 can be one of the possible answers, and we have taken to the maximum of all the possible answers.

Below is the implementation of the above approach: 

CPP




// C++ implementation of the approach
#include <bits/stdc++.h>
using namespace std;
 
// Function to calculate maximum GCD
int maxGCD(int N, int K)
{
 
    // Minimum possible sum for
    // K unique positive integers
    int minSum = (K * (K + 1)) / 2;
 
    // It is not possible to divide
    // N into K unique parts
    if (N < minSum)
        return -1;
 
    // All the factors greater than sqrt(N)
    // are complementary of the factors less
    // than sqrt(N)
    int i = sqrt(N);
    int res = 1;
    while (i >= 1) {
 
        // If i is a factor of N
        if (N % i == 0) {
            if (i >= minSum)
                res = max(res, N / i);
 
            if (N / i >= minSum)
                res = max(res, i);
        }
        i--;
    }
 
    return res;
}
 
// Driver code
int main()
{
    int N = 18, K = 3;
 
    cout << maxGCD(N, K);
 
    return 0;
}

Java




// Java implementation of the approach
import java.io.*;
import java.lang.Math;
 
class GFG
{
    // Function to calculate maximum GCD
    static int maxGCD(int N, int K)
    {
 
        // Minimum possible sum for
        // K unique positive integers
        int minSum = (K * (K + 1)) / 2;
 
        // It is not possible to divide
        // N into K unique parts
        if (N < minSum)
            return -1;
 
        // All the factors greater than sqrt(N)
        // are complementary of the factors less
        // than sqrt(N)
        int i = (int) Math.sqrt(N);
        int res = 1;
        while (i >= 1)
        {
 
            // If i is a factor of N
            if (N % i == 0)
            {
                if (i >= minSum)
                    res = Math.max(res, N / i);
 
                if (N / i >= minSum)
                    res = Math.max(res, i);
            }
            i--;
        }
 
        return res;
    }
 
    // Driver code
    public static void main (String[] args)
    {
        int N = 18, K = 3;
 
        System.out.println(maxGCD(N, K));
    }
}
 
// This code is contributed by ApurvaRaj

Python




# Python3 implementation of the approach
from math import sqrt,ceil,floor
 
# Function to calculate maximum GCD
def maxGCD(N, K):
 
    # Minimum possible sum for
    # K unique positive integers
    minSum = (K * (K + 1)) / 2
 
    # It is not possible to divide
    # N into K unique parts
    if (N < minSum):
        return -1
 
    # All the factors greater than sqrt(N)
    # are complementary of the factors less
    # than sqrt(N)
    i = ceil(sqrt(N))
    res = 1
    while (i >= 1):
 
        # If i is a factor of N
        if (N % i == 0):
            if (i >= minSum):
                res = max(res, N / i)
 
            if (N / i >= minSum):
                res = max(res, i)
 
        i-=1
 
    return res
 
# Driver code
 
N = 18
K = 3
 
print(maxGCD(N, K))
 
# This code is contributed by mohit kumar 29

C#




// C# implementation of the approach
using System;
 
class GFG
{
    // Function to calculate maximum GCD
    static int maxGCD(int N, int K)
    {
 
        // Minimum possible sum for
        // K unique positive integers
        int minSum = (K * (K + 1)) / 2;
 
        // It is not possible to divide
        // N into K unique parts
        if (N < minSum)
            return -1;
 
        // All the factors greater than sqrt(N)
        // are complementary of the factors less
        // than sqrt(N)
        int i = (int) Math.Sqrt(N);
        int res = 1;
        while (i >= 1)
        {
 
            // If i is a factor of N
            if (N % i == 0)
            {
                if (i >= minSum)
                    res = Math.Max(res, N / i);
 
                if (N / i >= minSum)
                    res = Math.Max(res, i);
            }
            i--;
        }
        return res;
    }
 
    // Driver code
    public static void Main()
    {
        int N = 18, K = 3;
 
        Console.WriteLine(maxGCD(N, K));
    }
}
 
// This code is contributed by AnkitRai01

Javascript




<script>
// javascript implementation of the approach
 
    // Function to calculate maximum GCD
    function maxGCD(N , K) {
 
        // Minimum possible sum for
        // K unique positive integers
        var minSum = (K * (K + 1)) / 2;
 
        // It is not possible to divide
        // N into K unique parts
        if (N < minSum)
            return -1;
 
        // All the factors greater than sqrt(N)
        // are complementary of the factors less
        // than sqrt(N)
        var i = parseInt( Math.sqrt(N));
        var res = 1;
        while (i >= 1) {
 
            // If i is a factor of N
            if (N % i == 0) {
                if (i >= minSum)
                    res = Math.max(res, N / i);
 
                if (N / i >= minSum)
                    res = Math.max(res, i);
            }
            i--;
        }
 
        return res;
    }
 
    // Driver code
     
        var N = 18, K = 3;
 
        document.write(maxGCD(N, K));
 
// This code contributed by Rajput-Ji
</script>

Output: 

3

 

Time Complexity: O(N1/2)

Auxiliary Space: O(1)


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