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

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 minimal of them.

Below is the implementation of the above approach.

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// 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;
}
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// 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
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# 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 //
  
            # Check for gcd > 1 
            if math.gcd(m, i) > 1
                return
  
            # 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
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// 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
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<?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.
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Output:
2

Time Complexity: O(sqrt N * log max(N, M))




Striver(underscore)79 at Codechef and codeforces D

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