Primality Test | Set 1 (Introduction and School Method)

Difficulty Level : Easy
Last Updated : 17 Mar, 2021

Given a positive integer, check if the number is prime or not. A prime is a natural number greater than 1 that has no positive divisors other than 1 and itself. Examples of first few prime numbers are {2, 3, 5,
Examples :

Input:  n = 11
Output: true

Input:  n = 15
Output: false

Input:  n = 1
Output: false

School Method
A simple solution is to iterate through all numbers from 2 to n-1 and for every number check if it divides n. If we find any number that divides, we return false.
Below is the implementation of this method.

C++

// A school method based C++ program to check if a
// number is prime
#include <bits/stdc++.h>
using namespace std;
 
bool isPrime(int n)
{
    // Corner case
    if (n <= 1)  return false;
 
    // Check from 2 to n-1
    for (int i=2; i<n; i++)
        if (n%i == 0)
            return false;
 
    return true;
}
 
// Driver Program to test above function
int main()
{
    isPrime(11)?  cout << " true\n": cout << " false\n";
    isPrime(15)?  cout << " true\n": cout << " false\n";
    return 0;
}

Java

// A school method based JAVA program
// to check if a number is prime
class GFG {
     
    static boolean isPrime(int n)
    {
        // Corner case
        if (n <= 1) return false;
     
        // Check from 2 to n-1
        for (int i = 2; i < n; i++)
            if (n % i == 0)
                return false;
     
        return true;
    }
     
    // Driver Program
    public static void main(String args[])
    {
        if(isPrime(11))
            System.out.println(" true");
        else
            System.out.println(" false");
        if(isPrime(15))
            System.out.println(" true");
        else
            System.out.println(" false");
         
    }
}
 
// This code is contributed
// by Nikita Tiwari.

Python3

# A school method based Python3
# program to check if a number
# is prime
 
def isPrime(n):
 
    # Corner case
    if n <= 1:
        return False
 
    # Check from 2 to n-1
    for i in range(2, n):
        if n % i == 0:
            return False;
 
    return True
 
# Driver Program to test above function
print("true") if isPrime(11) else print("false")
print("true") if isPrime(14) else print("false")
 
# This code is contributed by Smitha Dinesh Semwal

C#

// A optimized school method based C#
// program to check if a number is prime
using System;
 
namespace prime
{
    public class GFG
    {    
        public static bool isprime(int n)
        {
            // Corner cases
            if (n <= 1) return false;
             
            for (int i = 2; i < n; i++)
                if (n % i == 0)
                return false;
  
            return true;
        }
         
        // Driver program
        public static void Main()
        {
            if (isprime(11)) Console.WriteLine("true");
            else Console.WriteLine("false");
             
            if (isprime(15)) Console.WriteLine("true");
            else Console.WriteLine("false");
        }
    }
}
 
// This code is contributed by Sam007

PHP

<?php
// A school method based PHP
// program to check if a number
// is prime
 
function isPrime($n)
{
    // Corner case
    if ($n <= 1) return false;
 
    // Check from 2 to n-1
    for ($i = 2; $i < $n; $i++)
        if ($n % $i == 0)
            return false;
 
    return true;
}
 
// Driver Code
$tet = isPrime(11) ? " true\n" :
                     " false\n";
echo $tet;
$tet = isPrime(15) ? " true\n" :
                     " false\n";
echo $tet;
 
// This code is contributed by m_kit
?>

Javascript

<script>
 
// A school method based Javascript program to check if a
// number is prime
function isPrime(n)
{
    // Corner case
    if (n <= 1) return false;
 
    // Check from 2 to n-1
    for (let i = 2; i < n; i++)
        if (n % i == 0)
            return false;
    return true;
}
 
// Driver Program to test above function
    isPrime(11)? document.write(" true" + "<br>"): document.write(" false" + "<br>");
    isPrime(15)? document.write(" true" + "<br>"): document.write(" false" + "<br>");
 
// This code is contributed by Mayank Tyagi
 
</script>

Output :

true
false

Time complexity of this solution is O(n)
Optimized School Method
We can do following optimizations:

Instead of checking till n, we can check till √n because a larger factor of n must be a multiple of smaller factor that has been already checked.
The algorithm can be improved further by observing that all primes are of the form 6k ± 1, with the exception of 2 and 3. This is because all integers can be expressed as (6k + i) for some integer k and for i = -1, 0, 1, 2, 3, or 4; 2 divides (6k + 0), (6k + 2), (6k + 4); and 3 divides (6k + 3). So a more efficient method is to test if n is divisible by 2 or 3, then to check through all the numbers of form 6k ± 1. (Source: wikipedia)

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