Smallest divisor D of N such that gcd(D, M) is greater than 1

Namachila November 19, 2025

0 0 4 minutes read

Given two positive integers N and M., The task is to find the smallest divisor D of N such that gcd(D, M) > 1. If there are no such divisors, then print -1.
Examples:

Input: N = 8, M = 10
Output: 2
Input: N = 8, M = 1
Output: -1

A naive approach is to iterate for every factor and calculate the gcd of the factor and M. If it exceeds M, then we have the answer.
Time Complexity: O(N * log max(N, M))
An efficient approach is to iterate till sqrt(n) and check for gcd(i, m). If gcd(i, m) > 1, then we print and break it, else we check for gcd(n/i, m) and store the minimum of them.
Below is the implementation of the above approach.

C++

// C++ implementation of the above approach
#include <bits/stdc++.h>
using namespace std;
 
// Function to find the minimum divisor
int findMinimum(int n, int m)
{
    int mini = m;
 
    // Iterate for all factors of N
    for (int i = 1; i * i <= n; i++) {
        if (n % i == 0) {
            int sec = n / i;
 
            // Check for gcd > 1
            if (__gcd(m, i) > 1) {
                return i;
            }
 
            // Check for gcd > 1
            else if (__gcd(sec, m) > 1) {
                mini = min(sec, mini);
            }
        }
    }
 
    // If gcd is m itself
    if (mini == m)
        return -1;
    else
        return mini;
}
// Drivers code
int main()
{
    int n = 8, m = 10;
    cout << findMinimum(n, m);
    return 0;
}

Java

// Java implementation of the above approach
class GFG
{
 
static int __gcd(int a, int b) 
{ 
    if (b == 0) 
        return a; 
    return __gcd(b, a % b); 
     
} 
 
// Function to find the minimum divisor
static int findMinimum(int n, int m)
{
    int mini = m;
 
    // Iterate for all factors of N
    for (int i = 1; i * i <= n; i++)
    {
        if (n % i == 0)
        {
            int sec = n / i;
 
            // Check for gcd > 1
            if (__gcd(m, i) > 1) 
            {
                return i;
            }
 
            // Check for gcd > 1
            else if (__gcd(sec, m) > 1)
            {
                mini = Math.min(sec, mini);
            }
        }
    }
 
    // If gcd is m itself
    if (mini == m)
        return -1;
    else
        return mini;
}
 
// Driver code
public static void main (String[] args) 
{
    int n = 8, m = 10;
    System.out.println(findMinimum(n, m));
}
}
 
// This code is contributed by chandan_jnu

Python3

# Python3 implementation of the above approach
import math
 
# Function to find the minimum divisor 
def findMinimum(n, m): 
 
    mini, i = m, 1
     
    # Iterate for all factors of N 
    while i * i <= n: 
        if n % i == 0: 
            sec = n // i 
 
            # Check for gcd > 1 
            if math.gcd(m, i) > 1: 
                return i 
 
            # Check for gcd > 1 
            elif math.gcd(sec, m) > 1: 
                mini = min(sec, mini) 
             
        i += 1
 
    # If gcd is m itself 
    if mini == m:
        return -1
    else:
        return mini 
 
# Drivers code 
if __name__ == "__main__": 
 
    n, m = 8, 10
    print(findMinimum(n, m)) 
 
# This code is contributed by Rituraj Jain

C#

// C# implementation of the above approach
using System;
 
class GFG
{
 
static int __gcd(int a, int b) 
{ 
    if (b == 0) 
        return a; 
    return __gcd(b, a % b); 
     
} 
 
// Function to find the minimum divisor
static int findMinimum(int n, int m)
{
    int mini = m;
 
    // Iterate for all factors of N
    for (int i = 1; i * i <= n; i++)
    {
        if (n % i == 0)
        {
            int sec = n / i;
 
            // Check for gcd > 1
            if (__gcd(m, i) > 1) 
            {
                return i;
            }
 
            // Check for gcd > 1
            else if (__gcd(sec, m) > 1)
            {
                mini = Math.Min(sec, mini);
            }
        }
    }
 
    // If gcd is m itself
    if (mini == m)
        return -1;
    else
        return mini;
}
 
// Driver code
static void Main()
{
    int n = 8, m = 10;
    Console.WriteLine(findMinimum(n, m));
}
}
 
// This code is contributed by chandan_jnu

PHP

<?php
// PHP implementation of the above approach
function __gcd($a, $b) 
{ 
    if ($b == 0) 
        return $a; 
    return __gcd($b, $a % $b); 
     
} 
 
// Function to find the minimum divisor
function findMinimum($n, $m)
{
    $mini = $m;
 
    // Iterate for all factors of N
    for ($i = 1; $i * $i <= $n; $i++)
    {
        if ($n % $i == 0)
        {
            $sec = $n / $i;
 
            // Check for gcd > 1
            if (__gcd($m, $i) > 1) 
            {
                return $i;
            }
 
            // Check for gcd > 1
            else if (__gcd($sec, $m) > 1)
            {
                $mini = min($sec, $mini);
            }
        }
    }
 
    // If gcd is m itself
    if ($mini == $m)
        return -1;
    else
        return $mini;
}
 
// Driver code
$n = 8; $m = 10;
echo(findMinimum($n, $m));
 
// This code is contributed by Code_Mech.

Javascript

<script>
// javascript implementation of the above approach    
function __gcd(a , b) {
        if (b == 0)
            return a;
        return __gcd(b, a % b);
 
    }
 
    // Function to find the minimum divisor
    function findMinimum(n , m) {
        var mini = m;
 
        // Iterate for all factors of N
        for (var i = 1; i * i <= n; i++) {
            if (n % i == 0) {
                var sec = n / i;
 
                // Check for gcd > 1
                if (__gcd(m, i) > 1) {
                    return i;
                }
 
                // Check for gcd > 1
                else if (__gcd(sec, m) > 1) {
                    mini = Math.min(sec, mini);
                }
            }
        }
 
        // If gcd is m itself
        if (mini == m)
            return -1;
        else
            return mini;
    }
 
    // Driver code
     
        var n = 8, m = 10;
        document.write(findMinimum(n, m));
 
// This code is contributed by todaysgaurav
</script>

Output:

Time Complexity: O(sqrt(N) * log max(N, M)), as we are using a loop to traverse sqrt(N) times and we are using the inbuilt GCD function in each traversal which costs logN time. Where N and M are the two integers provided.
Auxiliary Space: O(1), as we are not using any extra space.

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