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Count of numbers up to N having at least one prime factor common with N
  • Last Updated : 31 May, 2021

Given an integer N, the task is to count the number of integers from the range [1, N] having at least one in common prime factor with N other than 1.

Examples:

Input: N = 5
Output: 1
Explanation: 
Since 5 is prime. Therefore, there is no other number which is at most N, that has at least one common factor with N except N itself. Therefore, the count is 1.

Input: N = 12
Output: 8

Naive Approach: The simplest approach to solve the given problem is to iterate through all the numbers over the range [1, N], and for every element, if the GCD of each number with N is greater than 1, then increment the count. Otherwise, check for the next number. After checking all the numbers, we print the total count obtained.



Below is the implementation of the above approach:

C++




// C++ program for the above approach
 
#include <bits/stdc++.h>
using namespace std;
 
// Function to count all the numbers
// in the range [1, N] having common
// factor with N other than 1
int countNumbers(int N)
{
    // Stores the count of numbers
    // having more than 1 factor with N
    int count = 0;
 
    // Iterate over the range [1, N]
    for (int i = 1; i <= N; i++) {
 
        // If gcd is not 1 then
        // increment the count
        if (__gcd(i, N) != 1)
            count++;
    }
 
    // Print the resultant count
    cout << count;
}
 
// Driver Code
int main()
{
    int N = 5;
    countNumbers(N);
 
    return 0;
}

Java




// Java program for the above approach
import java.util.*;
 
class GFG{
 
// Function to count all the numbers
// in the range [1, N] having common
// factor with N other than 1
static void countNumbers(int N)
{
     
    // Stores the count of numbers
    // having more than 1 factor with N
    int count = 0;
 
    // Iterate over the range [1, N]
    for(int i = 1; i <= N; i++)
    {
         
        // If gcd is not 1 then
        // increment the count
        if (__gcd(i, N) != 1)
            count++;
    }
 
    // Print the resultant count
    System.out.print(count);
}
 
static int __gcd(int a, int b)
{
    return b == 0 ? a : __gcd(b, a % b);
}
 
// Driver Code
public static void main(String[] args)
{
    int N = 5;
    countNumbers(N);
}
}
 
// This code is contributed by shikhasingrajput

Python3




# Python 3 program for the above approach
import math
 
# Function to count all the numbers
# in the range [1, N] having common
# factor with N other than 1
def countNumbers(N):
 
    # Stores the count of numbers
    # having more than 1 factor with N
    count = 0
 
    # Iterate over the range [1, N]
    for i in range(1, N + 1):
 
        # If gcd is not 1 then
        # increment the count
        if (math.gcd(i, N) != 1):
            count += 1
 
    # Print the resultant count
    print(count)
 
 
# Driver Code
if __name__ == "__main__":
 
    N = 5
    countNumbers(N)
 
    # This code is contributed by ukasp.

C#




// C# program for the above approach
using System;
 
public class GFG{
 
// Function to count all the numbers
// in the range [1, N] having common
// factor with N other than 1
static void countNumbers(int N)
{
     
    // Stores the count of numbers
    // having more than 1 factor with N
    int count = 0;
 
    // Iterate over the range [1, N]
    for(int i = 1; i <= N; i++)
    {
         
        // If gcd is not 1 then
        // increment the count
        if (__gcd(i, N) != 1)
            count++;
    }
 
    // Print the resultant count
    Console.Write(count);
}
 
static int __gcd(int a, int b)
{
    return b == 0 ? a : __gcd(b, a % b);
}
 
// Driver Code
public static void Main(String[] args)
{
    int N = 5;
    countNumbers(N);
}
}
 
  
 
// This code is contributed by 29AjayKumar

Javascript




<script>
// javascript program for the above approach
 
// Function to count all the numbers
// in the range [1, N] having common
// factor with N other than 1
function countNumbers(N)
{
     
    // Stores the count of numbers
    // having more than 1 factor with N
    var count = 0;
 
    // Iterate over the range [1, N]
    for(var i = 1; i <= N; i++)
    {
         
        // If gcd is not 1 then
        // increment the count
        if (__gcd(i, N) != 1)
            count++;
    }
 
    // Prvar the resultant count
    document.write(count);
}
 
function __gcd(a , b)
{
    return b == 0 ? a : __gcd(b, a % b);
}
 
// Driver Code
var N = 5;
countNumbers(N);
 
// This code is contributed by 29AjayKumar
</script>
Output: 
1

 

Time Complexity: O(N*log N)
Auxiliary Space: O(1)

Efficient Approach: The above approach can also be optimized by using Euler’s totient function which gives the count of numbers less than N having no common prime factor with N. Follow the steps below to solve the problem:

  • Initialize a variable, say count that stores the numbers having common prime factors other than 1.
  • Define a function, phi() to calculate the value of Euler’s totient function which represents the count of integers less than N having no common factor with N.
  • After completing the above steps, print the value of (N – phi(N) as the resultant count.

Below is the implementation of the above approach:

C++




// C++ program for the above approach
 
#include <bits/stdc++.h>
using namespace std;
 
// Function to calculate the value of
// Euler's totient function
int phi(int N)
{
    // Initialize result with N
    int result = N;
 
    // Find all prime factors of N
    // and subtract their multiples
    for (int p = 2; p * p <= N; ++p) {
 
        // Check if p is a prime factor
        if (N % p == 0) {
 
            // If found to be true,
            // then update N and result
            while (N % p == 0)
                N /= p;
 
            result -= result / p;
        }
    }
 
    // If N has a prime factor greater
    // than sqrt(N), then there can be
    // at-most one such prime factor
    if (N > 1)
        result -= result / N;
 
    return result;
}
 
// Function to count all the numbers
// in the range [1, N] having common
// factor with N other than 1
int countNumbers(int N)
{
    // Stores the resultant count
    int count = N - phi(N);
 
    // Print the count
    cout << count;
}
 
// Driver Code
int main()
{
    int N = 5;
    countNumbers(N);
 
    return 0;
}

Java




// Java program for the above approach
import java.util.*;
class GFG
{
     
 // Function to calculate the value of
// Euler's totient function
static int phi(int N)
{
   
    // Initialize result with N
    int result = N;
 
    // Find all prime factors of N
    // and subtract their multiples
    for (int p = 2; p * p <= N; ++p)
    {
 
        // Check if p is a prime factor
        if (N % p == 0) {
 
            // If found to be true,
            // then update N and result
            while (N % p == 0)
                N /= p;
 
            result -= result / p;
        }
    }
 
    // If N has a prime factor greater
    // than sqrt(N), then there can be
    // at-most one such prime factor
    if (N > 1)
        result -= result / N;
 
    return result;
}
 
// Function to count all the numbers
// in the range [1, N] having common
// factor with N other than 1
static void countNumbers(int N)
{
   
    // Stores the resultant count
    int count = N - phi(N);
 
    // Print the count
   System.out.print(count);
}
 
  // Driver code
public static void main (String[] args)
{
     int N = 5;
    countNumbers(N);
}
}
 
// This code is contributed by offbeat.

C#




// C# program for the above approach
using System;
 
public class GFG
{
     
 // Function to calculate the value of
// Euler's totient function
static int phi(int N)
{
   
    // Initialize result with N
    int result = N;
 
    // Find all prime factors of N
    // and subtract their multiples
    for (int p = 2; p * p <= N; ++p)
    {
 
        // Check if p is a prime factor
        if (N % p == 0) {
 
            // If found to be true,
            // then update N and result
            while (N % p == 0)
                N /= p;
 
            result -= result / p;
        }
    }
 
    // If N has a prime factor greater
    // than sqrt(N), then there can be
    // at-most one such prime factor
    if (N > 1)
        result -= result / N;
 
    return result;
}
 
// Function to count all the numbers
// in the range [1, N] having common
// factor with N other than 1
static void countNumbers(int N)
{
   
    // Stores the resultant count
    int count = N - phi(N);
 
    // Print the count
   Console.Write(count);
}
 
  // Driver code
public static void Main(String[] args)
{
     int N = 5;
    countNumbers(N);
}
}
 
// This code contributed by shikhasingrajput

Javascript




<script>
 
// Javascript implementation
// for the above approach
 
// Function to calculate the value of
// Euler's totient function
function phi(N)
{
    
    // Initialize result with N
    let result = N;
  
    // Find all prime factors of N
    // and subtract their multiples
    for (let p = 2; p * p <= N; ++p)
    {
  
        // Check if p is a prime factor
        if (N % p == 0) {
  
            // If found to be true,
            // then update N and result
            while (N % p == 0)
                N /= p;
  
            result -= result / p;
        }
    }
  
    // If N has a prime factor greater
    // than sqrt(N), then there can be
    // at-most one such prime factor
    if (N > 1)
        result -= result / N;
  
    return result;
}
  
// Function to count all the numbers
// in the range [1, N] having common
// factor with N other than 1
function countNumbers(N)
{
    
    // Stores the resultant count
    let count = N - phi(N);
  
    // Prlet the count
       document.write(count);
}
 
// Driver Code
     
    let N = 5;
    countNumbers(N);
 
// This code is contributed by susmitakundugoaldanga.
</script>
Output: 
1

 

Time Complexity: O(N1/2)
Auxiliary Space: O(1)

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