Largest palindrome not exceeding N which can be expressed as product of two 3-digit numbers

Given a positive integer N, the task is to find the largest palindromic number less than N and it can be expressed as the product of two 3-digit numbers.

Examples:

Input: N = 101110
Output: 101101
Explanation: The number 101101 ( = 143 * 707) is the largest palindromic number possible satisfying the conditions.

Input: N = 800000
Output: 793397
Explanation: The number 793397 ( = 869 × 913) is the largest palindromic number possible satisfying the conditions.

Naive Approach: The simplest generate all possible pairs from the range [100, 999] and for each pair, check if their product is a palindrome or not and is less than N or not. Print the maximum of all such products obtained as the required answer.

Time Complexity: O(N * 900²)
Auxiliary Space: O(1)

Alternate Approach: The problem can also be solved based on the observation that all the multiples of 11 are palindromes. Therefore, iterate over the range [100, 999] and for every value in the range, iterate over the multiples of 11 in the range [121, 999], and check for the required condition in each iteration. Follow the steps below to solve the problem:

• Initialize a variable, say num, to store the largest palindromic number satisfying the given conditions.
• Iterate over the range [100, 999] using a variable, say i, and perform the following steps: 
  • Iterate over the range [121, 999] using a variable, say j in multiples of 11.
    • Store the product of i and j in string x.
    • If the value of X is less than N and X is a palindrome, then update the value of num if X > num.
    • Otherwise, keep iterating for the next pair of integers.
• After the above steps, print the value of num.

Below is the implementation of the above approach:

C++

// C++ program for the above approach
#include <bits/stdc++.h>
using namespace std;

// Function to find the largest palindrome
// not exceeding N which can be expressed
// as the product of two 3-digit numbers
void palindrome_prod(string N){

      // Stores all palindromes
    vector<int> palindrome_list;

    for (int i = 101; i < 1000; i++)
    {
        for (int j = 121; j < 1000;
             j += (i % 11 == 0) ? 1 : 11)
        {

            // Stores the product
            int n = i * j;
            string x = to_string(n);
            string y = x;
            reverse(y.begin(), y.end());

            // Check if X is palindrome
            if (x == y){

                  // Check n is less than N
                if (n < stoi(N)){

                    // If true, append it
                    // in the list
                    palindrome_list.push_back(i * j);
                     }
                  }
              }
          }

    // Print the largest palindrome
    cout << (*max_element(palindrome_list.begin(),
                          palindrome_list.end()));
}

// Driver Code
int main()
{

  string N = "101110";

  // Function Call
  palindrome_prod(N);

  return 0;
}

// This code is contributed by mohit kumar 29

Java

// Java program for the above approach
import java.util.*;

class GFG
{

// Function to find the largest palindrome
// not exceeding N which can be expressed
// as the product of two 3-digit numbers
static void palindrome_prod(String N){

      // Stores all palindromes
    Vector<Integer> palindrome_list = new Vector<Integer>();

    for (int i = 101; i < 1000; i++)
    {
        for (int j = 121; j < 1000;
             j += (i % 11 == 0) ? 1 : 11)
        {

            // Stores the product
            int n = i * j;
            String x = String.valueOf(n);
            String y = x;
            reverse(y);

            // Check if X is palindrome
            if (x == y){

                  // Check n is less than N
                if (n < Integer.valueOf(N)){

                    // If true, append it
                    // in the list
                    palindrome_list.add(i * j);
                     }
                  }
              }
          }

    // Print the largest palindrome
    System.out.print(Collections.max(palindrome_list));
}

static String reverse(String input)
{
    char[] a = input.toCharArray();
    int l, r = a.length - 1;
    for (l = 0; l < r; l++, r--)
    {
        char temp = a[l];
        a[l] = a[r];
        a[r] = temp;
    }
    return String.valueOf(a);
}

// Driver Code
public static void main(String[] args)
{
  String N = "101110";

  // Function Call
  palindrome_prod(N);
}
}

// This code is contributed by 29AjayKumar

Python3

# Python program for the above approach

# Function to find the largest palindrome
# not exceeding N which can be expressed
# as the product of two 3-digit numbers
def palindrome_prod(N):

      # Stores all palindromes
    palindrome_list = []

    for i in range(101, 1000):
        for j in range(121, 1000, (1 if i % 11 == 0 else 11)):

            # Stores the product
            n = i * j
            x = str(n)

            # Check if X is palindrome
            if x == x[::-1]:

                  # Check n is less than N
                if n < N:

                    # If true, append it
                    # in the list
                    palindrome_list.append(i * j)

    # Print the largest palindrome
    print(max(palindrome_list))

# Driver Code

N = 101110

# Function Call
palindrome_prod(N)

C#

// C# program for the above approach
using System;
using System.Collections.Generic;
using System.Linq;
class GFG
{

// Function to find the largest palindrome
// not exceeding N which can be expressed
// as the product of two 3-digit numbers
static void palindrome_prod(String N){

      // Stores all palindromes
    List<int> palindrome_list = new List<int>();

    for (int i = 101; i < 1000; i++)
    {
        for (int j = 121; j < 1000;
             j += (i % 11 == 0) ? 1 : 11)
        {

            // Stores the product
            int n = i * j;
            String x = String.Join("", n);
            String y = x;
            reverse(y);

            // Check if X is palindrome
            if (x == y)
            {

                  // Check n is less than N
                if (n < Int32.Parse(N))
                {

                    // If true, append it
                    // in the list
                    palindrome_list.Add(i * j);
                }
            }
        }
    }

    // Print the largest palindrome
    Console.Write(palindrome_list.Max());
}

static String reverse(String input)
{
    char[] a = input.ToCharArray();
    int l, r = a.Length - 1;
    for (l = 0; l < r; l++, r--)
    {
        char temp = a[l];
        a[l] = a[r];
        a[r] = temp;
    }
    return String.Join("", a);
}

// Driver Code
public static void Main(String[] args)
{
  String N = "101110";

  // Function Call
  palindrome_prod(N);
}
}

// This code is contributed by 29AjayKumar

Javascript

<script>

// Javascript program for the above approach

// Function to find the largest palindrome
// not exceeding N which can be expressed
// as the product of two 3-digit numbers
function palindrome_prod(N){

    // Stores all palindromes
    var palindrome_list = [];
    var i,j;
    for (i = 101; i < 1000; i++)
    {
        for (j = 121; j < 1000;
             j += (i % 11 == 0) ? 1 : 11)
        {

            // Stores the product
            var n = i * j;
            var x = n.toString();
            var y = x;
            y = y.split("").reverse().join("");

            // Check if X is palindrome
            if (x == y){

                // Check n is less than N
                if (n < Number(N)){

                    // If true, append it
                    // in the list
                    palindrome_list.push(i * j);
                }
             }
          }
       }

    // Print the largest palindrome
    document.write(Math.max.apply(null, palindrome_list));
}

// Driver Code
  var N = "101110";

  // Function Call
  palindrome_prod(N);

</script>

Output:

101101

Time Complexity: O(N*900²)
Auxiliary Space: O(1)