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Cyclic Number
  • Difficulty Level : Expert
  • Last Updated : 29 Apr, 2021

A cyclic number is an integer in which cyclic permutations of the digits are successive multiples of the number. The most widely known is the six-digit number 142857 (Please see below explanation given in examples).
The following trivial cases are typically excluded for Cyclic Numbers. 
 

  • Single digits, e.g.: 5
  • Repeated digits, e.g.: 555
  • Repeated cyclic numbers, e.g.: 142857142857

Given a number, check if it is cyclic or not.
Examples: 
 

Input : 142857
Output : Yes
Explanation
    142857 × 1 = 142857
    142857 × 2 = 285714
    142857 × 3 = 428571
    142857 × 4 = 571428
    142857 × 5 = 714285
    142857 × 6 = 857142 

 

We generate all cyclic permutations of the number and check if every permutation divides number of not. We also check for three conditions. If any of the three conditions is true, we return false.
 

C++




// Program to check if a number is cyclic.
#include <bits/stdc++.h>
using namespace std;
 
#define ull unsigned long long int
 
// Function to generate all cyclic permutations
// of a number
bool isCyclic(ull N)
{
    // Count digits and check if all
    // digits are same
    ull num = N;
    int count = 0;
    int digit = num % 10;
    bool allSame = true;
    while (num) {
        count++;
        if (num % 10 != digit)
            allSame = false;
        num = num / 10;
    }
 
    // If all digits are same, then
    // not considered cyclic.
    if (allSame == true)
        return false;
 
    // If counts of digits is even and
    // two halves are same, then the
    // number is not considered cyclic.
    if (count % 2 == 0) {
        ull halfPower = pow(10, count / 2);
        ull firstHalf = N % halfPower;
        ull secondHalf = N / halfPower;
        if (firstHalf == firstHalf && isCyclic(firstHalf))
            return false;
    }
 
    num = N;
    while (1) {
 
        // Following three lines generates a
        // circular pirmutation of a number.
        ull rem = num % 10;
        ull div = num / 10;
        num = (pow(10, count - 1)) * rem + div;
 
        // If all the permutations are checked
        // and we obtain original number exit
        // from loop.
        if (num == N)
            break;
 
        if (num % N != 0)
            return false;
    }
 
    return true;
}
 
// Driver Program
int main()
{
    ull N = 142857;
    if (isCyclic(N))
        cout << "Yes";
    else
        cout << "No";
    return 0;
}

Java




// Java Program to check if a number is cyclic
 
class GFG {
    // Function to generate all cyclic
    // permutations of a number
    static boolean isCyclic(long N)
    {
        // Count digits and check if all
        // digits are same
        long num = N;
        int count = 0;
        int digit = (int)(num % 10);
        boolean allSame = true;
        while (num > 0) {
            count++;
            if (num % 10 != digit)
                allSame = false;
            num = num / 10;
        }
 
        // If all digits are same, then
        // not considered cyclic.
        if (allSame == true)
            return false;
 
        // If counts of digits is even and
        // two halves are same, then the
        // number is not considered cyclic.
        if (count % 2 == 0) {
            long halfPower = (long)Math.pow(10, count / 2);
            long firstHalf = N % halfPower;
            long secondHalf = N / halfPower;
            if (firstHalf == firstHalf && isCyclic(firstHalf))
                return false;
        }
 
        num = N;
        while (true) {
            // Following three lines generates a
            // circular pirmutation of a number.
            long rem = num % 10;
            long div = num / 10;
            num = (long)(Math.pow(10, count - 1))
                      * rem
                  + div;
 
            // If all the permutations are checked
            // and we obtain original number exit
            // from loop.
            if (num == N)
                break;
 
            if (num % N != 0)
                return false;
        }
 
        return true;
    }
 
    // Driver code
    public static void main(String[] args)
    {
        long N = 142857;
        if (isCyclic(N))
            System.out.print("Yes");
        else
            System.out.print("No");
    }
}
 
// This code is contributed by Anant Agarwal.

Python3




# Program to check if
# a number is cyclic
# Function to generate
# all cyclic permutations
# of a number
def isCyclic(N):
 
    # Count digits and check if all
    # digits are same
    num = N
    count = 0
    digit =(num % 10)
    allSame = True
 
    while (num>0):
        count+= 1
        if (num % 10 != digit):
            allSame = False
        num = num // 10
     
  
    # If all digits are same, then
    # not considered cyclic.
    if (allSame == True):
        return False
  
    # If counts of digits is even and
    # two halves are same, then the
    # number is not considered cyclic.
    if (count % 2 == 0):
     
        halfPower = pow(10, count//2)
        firstHalf = N % halfPower
        secondHalf = N / halfPower
        if (firstHalf == firstHalf and
            isCyclic(firstHalf)):
            return False
     
  
    num = N
    while (True):
  
        # Following three lines
        # generates a
        # circular pirmutation
        # of a number.
        rem = num % 10
        div = num // 10
        num = pow(10, count - 1) * rem + div
  
        # If all the permutations
        # are checked
        # and we obtain original
        # number exit
        # from loop.
        if (num == N):
            break
  
        if (num % N != 0):
            return False
     
    return True
 
# Driver code
 
N = 142857
if (isCyclic(N)):
    print("Yes")
else:
    print("No")
 
# This code is contributed
# by Anant Agarwal.

C#




// C# Program to check if a number is cyclic
using System;
 
class GFG {
     
    // Function to generate all cyclic
    // permutations of a number
    static bool isCyclic(long N)
    {
         
        // Count digits and check if all
        // digits are same
        long num = N;
        int count = 0;
        int digit = (int)(num % 10);
        bool allSame = true;
        while (num > 0)
        {
            count++;
            if (num % 10 != digit)
                allSame = false;
            num = num / 10;
        }
 
        // If all digits are same, then
        // not considered cyclic.
        if (allSame == true)
            return false;
 
        // If counts of digits is even and
        // two halves are same, then the
        // number is not considered cyclic.
        if (count % 2 == 0) {
            long halfPower = (long)Math.Pow(10,
                                    count / 2);
                                     
            long firstHalf = N % halfPower;
             
            // long secondHalf = N / halfPower;
            if (firstHalf == firstHalf &&
                           isCyclic(firstHalf))
                return false;
        }
 
        num = N;
        while (true)
        {
             
            // Following three lines generates a
            // circular pirmutation of a number.
            long rem = num % 10;
            long div = num / 10;
            num = (long)(Math.Pow(10, count - 1))
                    * rem + div;
 
            // If all the permutations are checked
            // and we obtain original number exit
            // from loop.
            if (num == N)
                break;
 
            if (num % N != 0)
                return false;
        }
 
        return true;
    }
 
    // Driver code
    public static void Main()
    {
        long N = 142857;
         
        if (isCyclic(N))
            Console.WriteLine("Yes");
        else
            Console.WriteLine("No");
    }
}
 
// This code is contributed by vt_m.

PHP




<?php
// Program to check if a number is cyclic
 
// Function to generate all cyclic
// permutations of a number
function isCyclic($N)
{
    // Count digits and check if all
    // digits are same
    $num = $N;
    $count = 0;
    $digit = ($num % 10);
    $allSame = true;
 
    while ($num > 0)
    {
        $count += 1;
        if ($num % 10 != $digit)
        $allSame = false;
        $num = (int)($num / 10);
    }
 
    // If all digits are same, then
    // not considered cyclic.
    if ($allSame == true)
        return false;
 
    // If counts of digits is even and
    // two halves are same, then the
    // number is not considered cyclic.
    if ($count % 2 == 0)
    {
        $halfPower = pow(10, (int)($count / 2));
        $firstHalf = $N % $halfPower;
        $secondHalf = $N / $halfPower;
        if ($firstHalf == $firstHalf &&
                 isCyclic($firstHalf))
            return false;
    }
 
    $num = $N;
    while (true)
    {
 
        // Following three lines generates a
        // circular pirmutation of a number.
        $rem = $num % 10;
        $div = (int)($num / 10);
        $num = pow(10, $count - 1) * $rem + $div;
 
        // If all the permutations are checked
        // and we obtain original number, exit
        // from loop.
        if ($num == $N)
            break;
 
        if ($num % $N != 0)
            return false;
    }
    return true;
}
 
// Driver code
$N = 142857;
if (isCyclic($N))
    print("Yes");
else
    print("No");
 
// This code is contributed by mits
?>

Javascript




<script>
// Javascript Program to check if a number is cyclic
 
    // Function to generate all cyclic
    // permutations of a number
    function isCyclic(N)
    {
        // Count digits and check if all
        // digits are same
        let num = N;
        let count = 0;
        let digit = Math.floor(num % 10);
        let allSame = true;
        while (num > 0) {
            count++;
            if (num % 10 != digit)
                allSame = false;
            num = Math.floor(num / 10);
        }
   
        // If all digits are same, then
        // not considered cyclic.
        if (allSame == true)
            return false;
   
        // If counts of digits is even and
        // two halves are same, then the
        // number is not considered cyclic.
        if (count % 2 == 0) {
            let halfPower = Math.floor(Math.pow(10, count / 2));
            let firstHalf = Math.floor(N % halfPower);
            let secondHalf = Math.floor(N / halfPower);
            if (firstHalf == firstHalf && isCyclic(firstHalf))
                return false;
        }
   
        num = N;
        while (true) {
            // Following three lines generates a
            // circular pirmutation of a number.
            let rem = num % 10;
            let div = Math.floor(num / 10);
            num = Math.floor(Math.pow(10, count - 1))
                      * rem
                  + div;
   
            // If all the permutations are checked
            // and we obtain original number exit
            // from loop.
            if (num == N)
                break;
   
            if (num % N != 0)
                return false;
        }
   
        return true;
    }
 
// driver function
 
        let N = 142857;
        if (isCyclic(N))
            document.write("Yes");
        else
            document.write("No");
 
</script>

Output: 
 

Yes

Reference : 
https://en.wikipedia.org/wiki/Cyclic_number
This article is contributed by Ajay Puri. If you like GeeksforGeeks and would like to contribute, you can also write an article using write.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.
Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above..
 

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