Find the smallest number whose digits multiply to a given number n

Given a number ‘n’, find the smallest number ‘p’ such that if we multiply all digits of ‘p’, we get ‘n’. The result ‘p’ should have minimum two digits.

Examples:

Input:  n = 36
Output: p = 49 
// Note that 4*9 = 36 and 49 is the smallest such number

Input:  n = 100
Output: p = 455
// Note that 4*5*5 = 100 and 455 is the smallest such number

Input: n = 1
Output:p = 11
// Note that 1*1 = 1

Input: n = 13
Output: Not Possible


For a given n, following are the two cases to be considered.
Case 1: n < 10 When n is smaller than n, the output is always n+10. For example for n = 7, output is 17. For n = 9, output is 19.

Case 2: n >= 10 Find all factors of n which are between 2 and 9 (both inclusive). The idea is to start searching from 9 so that the number of digits in result are minimized. For example 9 is preferred over 33 and 8 is preferred over 24.
Store all found factors in an array. The array would contain digits in non-increasing order, so finally print the array in reverse order.

Following is the implementation of above concept.

C/C++

#include<stdio.h>
  
// Maximum number of digits in output
#define MAX 50
  
// prints the smallest number whose digits multiply to n
void findSmallest(int n)
{
    int i, j=0;
    int res[MAX]; // To sore digits of result in reverse order
  
    // Case 1: If number is smaller than 10
    if (n < 10)
    {
        printf("%d", n+10);
        return;
    }
  
    // Case 2: Start with 9 and try every possible digit
    for (i=9; i>1; i--)
    {
        // If current digit divides n, then store all
        // occurrences of current digit in res
        while (n%i == 0)
        {
            n = n/i;
            res[j] = i;
            j++;
        }
    }
  
    // If n could not be broken in form of digits (prime factors of n
    // are greater than 9)
    if (n > 10)
    {
        printf("Not possible");
        return;
    }
  
    // Print the result array in reverse order
    for (i=j-1; i>=0; i--)
        printf("%d", res[i]);
}
  
// Driver program to test above function
int main()
{
    findSmallest(7);
    printf("\n");
  
    findSmallest(36);
    printf("\n");
  
    findSmallest(13);
    printf("\n");
  
    findSmallest(100);
    return 0;
}

Java

// Java program to find the smallest number whose 
// digits multiply to a given number n
  
import java.io.*;
  
class Smallest
{
    // Function to prints the smallest number whose 
    // digits multiply to n
    static void findSmallest(int n)
    {
        int i, j=0;
        int MAX = 50;
        // To sore digits of result in reverse order
        int[] res = new int[MAX]; 
   
        // Case 1: If number is smaller than 10
        if (n < 10)
        {
            System.out.println(n+10);
            return;
        }
   
        // Case 2: Start with 9 and try every possible digit
        for (i=9; i>1; i--)
        {
            // If current digit divides n, then store all
            // occurrences of current digit in res
            while (n%i == 0)
            {
                n = n/i;
                res[j] = i;
                j++;
            }
        }
   
        // If n could not be broken in form of digits (prime factors of n
        // are greater than 9)
        if (n > 10)
        {
            System.out.println("Not possible");
            return;
        }
   
        // Print the result array in reverse order
        for (i=j-1; i>=0; i--)
            System.out.print(res[i]);
        System.out.println();
    }
      
    // Driver program
    public static void main (String[] args) 
    {
        findSmallest(7);
        findSmallest(36);
        findSmallest(13);
        findSmallest(100);
    }
}
  
// Contributed by Pramod Kumar

Python

# Python code to find the smallest number 
# whose digits multiply to give n
  
# function to print the smallest number whose 
# digits multiply to n
def findSmallest(n):
    # Case 1 - If the number is smaller than 10
    if n < 10:
        print n+10
        return
      
    # Case 2 - Start with 9 and try every possible digit
    res = [] # to sort digits
    for i in range(9,1,-1):
        # If current digit divides n, then store all
        # occurences of current digit in res
        while n % i == 0:
            n = n / i
            res.append(i)
      
    # If n could not be broken in form of digits
    # prime factors of  n are greater than 9
      
    if n > 10:
        print "Not Possible"
        return
          
    # Print the number from result array in reverse order
    n = res[len(res)-1]
    for i in range(len(res)-2,-1,-1):
        n = 10 * n + res[i]
    print n
      
# Driver Code to test above function
  
findSmallest(7)
  
findSmallest(36)
  
findSmallest(13)
  
findSmallest(100)
  
# This code is contributed by Harshit Agrawal

C#

// C# program to find the smallest number whose 
// digits multiply to a given number n
using System;
  
class GFG {
      
    // Function to prints the smallest number
    // whose digits multiply to n
    static void findSmallest(int n)
    {
          
        int i, j=0;
        int MAX = 50;
          
        // To sore digits of result in
        // reverse order
        int []res = new int[MAX]; 
  
        // Case 1: If number is smaller than 10
        if (n < 10)
        {
            Console.WriteLine(n + 10);
            return;
        }
  
        // Case 2: Start with 9 and try every
        // possible digit
        for (i = 9; i > 1; i--)
        {
              
            // If current digit divides n, then
            // store all occurrences of current
            // digit in res
            while (n % i == 0)
            {
                n = n / i;
                res[j] = i;
                j++;
            }
        }
  
        // If n could not be broken in form of
        // digits (prime factors of n
        // are greater than 9)
        if (n > 10)
        {
            Console.WriteLine("Not possible");
            return;
        }
  
        // Print the result array in reverse order
        for (i = j-1; i >= 0; i--)
            Console.Write(res[i]);
              
        Console.WriteLine();
    }
      
    // Driver program
    public static void Main () 
    {
        findSmallest(7);
        findSmallest(36);
        findSmallest(13);
        findSmallest(100);
    }
}
  
// This code is contributed by nitin mittal.

PHP

<?php
// PHP program to find the 
// smallest number whose 
// digits multiply to a 
// given number n prints the
// smallest number whose digits
// multiply to n
  
function findSmallest($n)
{
      
    // To sore digits of 
    // result in reverse order
    $i;
    $j = 0;
    $res
  
    // Case 1: If number is 
    // smaller than 10
    if ($n < 10)
    {
        echo $n + 10;
        return;
    }
  
    // Case 2: Start with 9 and 
    // try every possible digit
    for ($i = 9; $i > 1; $i--)
    {
          
        // If current digit divides 
        // n, then store all
        // occurrences of current 
        // digit in res
        while ($n % $i == 0)
        {
            $n = $n / $i;
            $res[$j] = $i;
            $j++;
        }
    }
  
    // If n could not be broken 
    // in form of digits 
    // (prime factors of n
    // are greater than 9)
    if ($n > 10)
    {
        echo "Not possible";
        return;
    }
  
    // Print the result 
    // array in reverse order
    for ($i = $j - 1; $i >= 0; $i--)
        echo $res[$i];
}
  
    // Driver Code
    findSmallest(7);
    echo "\n";
  
    findSmallest(36);
      
    echo "\n";
  
    findSmallest(13);
    echo "\n";
  
    findSmallest(100);
  
// This code is contributed by ajit
?>


Output:

17
49
Not possible
455 

This article is contributed by Ashish Bansal. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above



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Improved By : nitin mittal, jit_t