C++ Program To Check Whether a Number is Prime or not

Given a positive integer N. The task is to write a C++ program to check if the number is prime or not. 

Definition: 

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. The first few prime numbers are {2, 3, 5, 7, 11, ….}

The idea to solve this problem is to iterate through all the numbers starting from 2 to sqrt(N) using a for loop and for every number check if it divides N. If we find any number that divides, we return false. If we did not find any number between 2 and sqrt(N) which divides N then it means that N is prime and we will return True. 
Why did we choose sqrt(N)?
The reason is that the smallest and greater than one factor of a number cannot be more than the sqrt of N. And we stop as soon as we find a factor. For example, if N is 49, the smallest factor is 7. For 15, smallest factor is 3.
Below is the C++ program to check if a number is prime: 

C++

// C++ program to check if a
// number is prime
#include <iostream>
#include <math.h>
using namespace std;

int main()
{
    int n, i, flag = 1;

    // Ask user for input
    cout << "Enter a number: ";

    // Store input number in a variable
    cin >> n;

    // Iterate from 2 to sqrt(n)
    for (i = 2; i <= sqrt(n); i++) {

        // If n is divisible by any number between
        // 2 and n/2, it is not prime
        if (n % i == 0) {
            flag = 0;
            break;
        }
    }

    if (n <= 1)
        flag = 0;

    if (flag == 1) {
        cout << n << " is a prime number";
    }
    else {
        cout << n << " is not a prime number";
    }

    return 0;
}

// This code is contributed by shivanisinghss2110.

Output: 

Enter a number: 11
11 is a prime number

Time Complexity: O(n^1/2)
Auxiliary Space: O(1)

Method 2 : Optimized Approach using Wilsons theorem with O(N) complexity

If   ((n-1)! + 1) % n == 0  then  n is prime and else it is not prime

Example : 

Input : 11
Output : 11 is a prime number
Explanation : (11-1)! + 1 = 3628801
                       3628801 % 11 = 0 

Input : 8
Output : 8 is not prime number
Explanation : (8-1)! + 1 = 5041
                       5041 % 8 = 1 

Implementation of the above approach : 

C++

// C++ program to check whether number is prime or not
#include <iostream>
using namespace std;

int main()
{
    // code
    int n = 11;
    int m = n - 1;
    int factm = 1;
    // find factorial of n-1
    for (int i = 1; i <= m; i++) {
        factm *= i;
    }

    // add 1 to (n-1)!
    int factn = factm + 1;
    if (factn % n == 0) {
        // if remainder is 0
        cout << n << " is prime number";
    }
    else {
        cout << n << " is not prime";
    }
    return 0;
}
// this code is contributed by devendra solunke

Output

 11  is prime number

Time Complexity: O(N) complexity only for calculating factorial  of (n-1) checking it is 0 or 1 using % takes constant time
Auxiliary Space: O(1)