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.

`// 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"` `);` ` ` `}` `}` |

**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)
`// A optimized school method based Java`

`// program to check if a number is prime`

`import`

`java.io.*;`

`class`

`GFG {`

`static`

`boolean`

`isPrime(`

`int`

`n)`

`{`

`// Corner cases`

`if`

`(n <=`

`1`

`)`

`return`

`false`

`;`

`if`

`(n <=`

`3`

`)`

`return`

`true`

`;`

`// This is checked so that we can skip`

`// middle five numbers in below loop`

`if`

`(n %`

`2`

`==`

`0`

`|| n %`

`3`

`==`

`0`

`)`

`return`

`false`

`;`

`for`

`(`

`int`

`i =`

`5`

`; i * i <= n; i = i +`

`6`

`)`

`if`

`(n % i ==`

`0`

`|| n % (i +`

`2`

`) ==`

`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"`

`);`

`}`

`}`

**Output:**true false

Time complexity of this solution is O(√n)

Main Article : Primality Test | Set 1 (Introduction and School Method)

**References:**

https://en.wikipedia.org/wiki/Prime_number

http://www.cse.iitk.ac.in/users/manindra/presentations/FLTBasedTests.pdf

https://en.wikipedia.org/wiki/Primality_testThis article is contributed by

**Ajay**. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed aboveAttention reader! Don’t stop learning now. Get hold of all the important DSA concepts with the

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