# Wheel Factorization Algorithm

• Difficulty Level : Hard
• Last Updated : 19 May, 2021

Given a number N. The task is to check if the given number is Prime Number or not.
Examples:

Input: N = 987
Output: Not a Prime Number
Explanation:
As, 987 = 3*7*47. Therefore 987 is not a prime number.
Input: N = 67
Output: Prime Number

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Wheel Factorization Method:
Wheel Factorization is the improvement of the method Sieve of Eratosthenes. For wheel factorization, one starts from a small list of numbers, called the basis — generally the first few prime numbers, then one generates the list, called the wheel, of the integers that are coprime with all numbers of the basis. Then to find the smallest divisor of the number to be factorized, one divides it successively by the numbers on the basis, and in the wheel Let say we select basis as {2, 3, 5} and the numbers which are coprime to the basis are {7, 11, 13, 17, 19, 23, 29, 31} are set as the wheel
To understand it more, see the pattern in the above image that the numbers exhibit. The LCM of the first three Prime Numbers is 30. The numbers(less than 30) which are ending with 7, 1, and 3 and are not a multiple of 2, 3, and 5 and are always prime i.e {7, 11, 13, 17, 19, 23, 29}. Adding the no. 31 to this list and then if we add multiples of 30 to any of the numbers in the list, it gives us a Prime Number
Algorithm for Wheel Factorization Method:

```For the number N to be Prime or not:
bool isPrime(x) {
if (x < 2) {
return False;
}
for N in {2, 3, 5} {
return False;
}
for p= [0, sqrt(N)] such that p = p + 30: {
for c in [p+7, p+11, p+13, p+17, p+19, p+23, p+29, p+31] {
if c > sqrt(N)
break;
else if N % (c+p) = 0:
return False;
}
}
}
return True;
}```

Approach:
Following is the approach for the above algorithm:

1. For Primality Test of a given Number N, check if the given number is divisible by any of the number 2, 3, 5 or not.
2. If the number is not divisible by any of 2, 3, 5, then check if the number formed by adding multiples of 30 in the list [7, 11, 13, 17, 19, 23, 29, 31] divides the given number N or not. If Yes then the given number is not Prime Number, else it is a Prime Number.

Below is the implementation of the above approach:

## C++

 `// C++ program to check if the``// given number is prime using``// Wheel Factorization Method``#include "bits/stdc++.h"``using` `namespace` `std;` `// Function to check if a given``// number x is prime or not``void` `isPrime(``int` `N)``{``    ``bool` `isPrime = ``true``;``    ``// The Wheel for checking``    ``// prime number``    ``int` `arr = { 7, 11, 13, 17,``                   ``19, 23, 29, 31 };` `    ``// Base Case``    ``if` `(N < 2) {``        ``isPrime = ``false``;``    ``}` `    ``// Check for the number taken``    ``// as basis``    ``if` `(N % 2 == 0 || N % 3 == 0``        ``|| N % 5 == 0) {``        ``isPrime = ``false``;``    ``}` `    ``// Check for Wheel``    ``// Here i, acts as the layer``    ``// of the wheel``    ``for` `(``int` `i = 0; i < ``sqrt``(N); i += 30) {` `        ``// Check for the list of``        ``// Sieve in arr[]``        ``for` `(``int` `c : arr) {` `            ``// If number is greater``            ``// than sqrt(N) break``            ``if` `(c > ``sqrt``(N)) {``                ``break``;``            ``}` `            ``// Check if N is a multiple``            ``// of prime number in the``            ``// wheel``            ``else` `{``                ``if` `(N % (c + i) == 0) {``                    ``isPrime = ``false``;``                    ``break``;``                ``}``            ``}` `            ``// If at any iteration``            ``// isPrime is false,``            ``// break from the loop``            ``if` `(!isPrime)``                ``break``;``        ``}``    ``}` `    ``if` `(isPrime)``        ``cout << ``"Prime Number"``;``    ``else``        ``cout << ``"Not a Prime Number"``;``}` `// Driver's Code``int` `main()``{``    ``int` `N = 121;` `    ``// Function call for primality``    ``// check``    ``isPrime(N);``    ``return` `0;``}`

## Java

 `// Java program to check if the``// given number is prime using``// Wheel Factorization Method``import` `java.util.*;` `class` `GFG{` `// Function to check if a given``// number x is prime or not``static` `void` `isPrime(``int` `N)``{``    ``boolean` `isPrime = ``true``;``    ` `        ``// The Wheel for checking``    ``// prime number``    ``int` `[]arr = { ``7``, ``11``, ``13``, ``17``,``19``, ``23``, ``29``, ``31` `};` `    ``// Base Case``    ``if` `(N < ``2``) {``        ``isPrime = ``false``;``    ``}` `    ``// Check for the number taken``    ``// as basis``    ``if` `(N % ``2` `== ``0` `|| N % ``3` `== ``0``        ``|| N % ``5` `== ``0``) {``        ``isPrime = ``false``;``    ``}` `    ``// Check for Wheel``    ``// Here i, acts as the layer``    ``// of the wheel``    ``for` `(``int` `i = ``0``; i < Math.sqrt(N); i += ``30``) {` `        ``// Check for the list of``        ``// Sieve in arr[]``        ``for` `(``int` `c : arr) {` `            ``// If number is greater``            ``// than sqrt(N) break``            ``if` `(c > Math.sqrt(N)) {``                ``break``;``            ``}` `            ``// Check if N is a multiple``            ``// of prime number in the``            ``// wheel``            ``else` `{``                ``if` `(N % (c + i) == ``0``) {``                    ``isPrime = ``false``;``                    ``break``;``                ``}``            ``}` `            ``// If at any iteration``            ``// isPrime is false,``            ``// break from the loop``            ``if` `(!isPrime)``                ``break``;``        ``}``    ``}` `    ``if` `(isPrime)``        ``System.out.println(``"Prime Number"``);``    ``else``        ``System.out.println(``"Not a Prime Number"``);``}` `// Driver's Code``public` `static` `void` `main(String args[])``{``    ``int` `N = ``121``;` `    ``// Function call for primality``    ``// check``    ``isPrime(N);``}``}` `// This code is contributed by Surendra_Gangwar`

## C#

 `// C# program to check if the``// given number is prime using``// Wheel Factorization Method``using` `System;` `class` `GFG{` `// Function to check if a given``// number x is prime or not``static` `void` `isPrime(``int` `N)``{``    ``bool` `isPrime = ``true``;``    ` `     ``// The Wheel for checking``    ``// prime number``    ``int` `[]arr = { 7, 11, 13, 17,19, 23, 29, 31 };` `    ``// Base Case``    ``if` `(N < 2) {``        ``isPrime = ``false``;``    ``}` `    ``// Check for the number taken``    ``// as basis``    ``if` `(N % 2 == 0 || N % 3 == 0``        ``|| N % 5 == 0) {``        ``isPrime = ``false``;``    ``}` `    ``// Check for Wheel``    ``// Here i, acts as the layer``    ``// of the wheel``    ``for` `(``int` `i = 0; i < (``int``)Math.Sqrt(N); i += 30) {``    ` `        ``// Check for the list of``        ``// Sieve in arr[]``        ``foreach` `(``int` `c ``in` `arr) {` `            ``// If number is greater``            ``// than sqrt(N) break``            ``if` `(c > (``int``)Math.Sqrt(N)) {``                ``break``;``            ``}` `            ``// Check if N is a multiple``            ``// of prime number in the``            ``// wheel``            ``else` `{``                ``if` `(N % (c + i) == 0) {``                    ``isPrime = ``false``;``                    ``break``;``                ``}``            ``}` `            ``// If at any iteration``            ``// isPrime is false,``            ``// break from the loop``            ``if` `(!isPrime)``                ``break``;``        ``}``    ``}` `    ``if` `(isPrime)``        ``Console.WriteLine(``"Prime Number"``);``    ``else``        ``Console.WriteLine(``"Not a Prime Number"``);``}` `// Driver's Code``public` `static` `void` `Main(String []args)``{``    ``int` `N = 121;` `    ``// Function call for primality``    ``// check``    ``isPrime(N);``}``}` `// This code is contributed by Yash_R`

## Python3

 `# Python3  program to check if the``# given number is prime using``# Wheel Factorization Method``import` `math` `# Function to check if a given``# number x is prime or not``def` `isPrime( N):` `    ``isPrime ``=` `True``;``    ``# The Wheel for checking``    ``# prime number``    ``arr``=` `[ ``7``, ``11``, ``13``, ``17``,``                ``19``, ``23``, ``29``, ``31` `]` `    ``# Base Case``    ``if` `(N < ``2``) :``        ``isPrime ``=` `False``    ` `    ``# Check for the number taken``    ``# as basis``    ``if` `(N ``%` `2` `=``=` `0` `or` `N ``%` `3` `=``=` `0``        ``or` `N ``%` `5` `=``=` `0``):``        ``isPrime ``=` `False``    ` `    ``# Check for Wheel``    ``# Here i, acts as the layer``    ``# of the wheel``    ``for` `i ``in` `range``(``0``,``int``(math.sqrt(N)), ``30``) :` `        ``# Check for the list of``        ``# Sieve in arr[]``        ``for` `c ``in`  `arr:` `            ``# If number is greater``            ``# than sqrt(N) break``            ``if` `(c > ``int``(math.sqrt(N))):``                ``break``            ` `            ``# Check if N is a multiple``            ``# of prime number in the``            ``# wheel``            ``else` `:``                ``if` `(N ``%` `(c ``+` `i) ``=``=` `0``) :``                    ``isPrime ``=` `False``                    ``break` `            ``# If at any iteration``            ``# isPrime is false,``            ``# break from the loop``            ``if` `(``not` `isPrime):``                ``break` `    ``if` `(isPrime):``        ``print``(``"Prime Number"``)``    ``else``:``        ``print``(``"Not a Prime Number"``)` `# Driver's Code``if` `__name__ ``=``=` `"__main__"``:``    ``N ``=` `121` `    ``# Function call for primality``    ``# check``    ``isPrime(N)` `# This code is contributed by chitranayal`

## Javascript

 ``
Output:
`Not a Prime Number`

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