Check if Array elements can be made equal by adding unit digit to the number

Given an array arr[] of N integers, the task is to check whether it is possible to make all the array elements identical by applying the following option any number of times:

• Choose any number and add its unit digit to it.

Examples:

Input: arr[] = {1, 2, 4, 8, 24}
Output: Possible
Explanation: 
For 1: 1 -> 2 -> 4 -> 8 -> 16- > 22 -> 24
For 2: 2 -> 4 -> 8 -> 16 -> 22 -> 24
For 4: 4 -> 8 -> 16 -> 22 -> 24
For 8: 16- > 22 -> 24
For 24: 24 (it's already 24, so, no need to change)

Input: arr[] = {5, 10}
Output: Possible

Approach: The problem can be solved based on the following observation.

Any number can be converted to have either 0 or 2 as its unit digit by repetitively adding its unit digit to itself at most 10 times. 

The idea is to 

• Initially make the unit digit of every number either 2 or 0.
• Then for the numbers that have 0 as their unit digit, all of those numbers must be same(because x+0 = 0), and 
• For every number with 2 as the unit digit, the difference between them must be multiple of 20 so that they can be made equal.

Follow the steps to solve this problem:

• Add the unit digit of each number to itself until its unit digit becomes 2 or 0.
• For every number that has 0 as a unit digit, check whether all the numbers are equal.
• For numbers with 2 as the unit digit, the difference between them must be a multiple of 20.

Below is the implementation of the above approach:

// C++ code to implement the approach

#include <bits/stdc++.h>
using namespace std;

// Function to check whether the elements
// of array can be equal or not
string solve(int* arr, int n)
{
    // Initialize flag as 1
    bool flag = 1;

    // Make unit digit of every element of array
    // with either 0 or 2
    for (int i = 0; i < n; i++) {
        while (arr[i] % 10 != 0 && arr[i] % 10 != 2)
            arr[i] += arr[i] % 10;
    }

    // Check if it is possible to make
    // the array elements identical or not
    for (int i = 0; i < n; i++) {

        // Elements for 0 as unit digit
        if ((arr[0] % 10 == 0) && (arr[i] != arr[0])) {
            flag = 0;
            break;
        }

        // Elements for 2 as unit digit
        if ((arr[i] - arr[0]) % 20) {
            flag = 0;
            break;
        }
    }

    if (flag)
        return "Possible";
    else
        return "Impossible";
}

// Driver Code
int main()
{
    int arr[] = { 1, 2, 4, 8, 24 };
    int N = sizeof(arr) / sizeof(arr[0]);

    // Function call
    cout << solve(arr, N);
    return 0;
}

// Java code for the above approach
import java.io.*;

class GFG
{

// Function to check whether the elements
// of array can be equal or not
static String solve(int[] arr, int n)
{
    // Initialize flag as 1
    boolean flag = true;

    // Make unit digit of every element of array
    // with either 0 or 2
    for (int i = 0; i < n; i++) {
        while (arr[i] % 10 != 0 && arr[i] % 10 != 2)
            arr[i] += arr[i] % 10;
    }

    // Check if it is possible to make
    // the array elements identical or not
    for (int i = 0; i < n; i++) {

        // Elements for 0 as unit digit
        if ((arr[0] % 10 == 0) && (arr[i] != arr[0])) {
            flag = false;
            break;
        }

        // Elements for 2 as unit digit
        if (((arr[i] - arr[0]) % 20) != 0) {
            flag = false;
            break;
        }
    }

    if (flag )
        return "Possible";
    else
        return "Impossible";
}

    // Driver code
    public static void main(String[] args)
    {
        int arr[] = { 1, 2, 4, 8, 24 };
    int N = arr.length;

    // Function call
    System.out.println( solve(arr, N));
    }
}

 // This code is contributed by sanjoy_62.

# Python3 code for the above approach

# Function to check whether the elements of array can
# be equal or not
def solve(arr, n):
  
      # Initialize flag as true
    flag = True

    # Make unit digit of every element of array with
    # either 0 or 2
    # for (int i = 0; i < n; i++) {
    for i in range(0, n):
        while ((arr[i] % 10 != 0) and (arr[i] % 10 != 2)):
            arr[i] += arr[i] % 10

            # Check if it is possible to make the array
            # elements identical or not
            # for (int i = 0; i < n; i++) {
    for i in range(0, n):
              # Elements for 0 as unit digit
        if ((arr[0] % 10) == 0 and (arr[i] != arr[0])):
            flag = False
            break

            # Elements for 2 as unit digit
        if ((arr[i] - arr[0]) % 20 != 0):

            flag = False
            break

    if (flag):
        return "Possible"
    else:
        return "Impossible"

# Driver Code
arr = [1, 2, 4, 8, 24]
N = len(arr)

# Function call
ans = solve(arr, N)
print(ans)

# This code is contributed by akashish.

// C# code for the above approach

using System;

public class GFG {

    // Function to check whether the elements of array can
    // be equal or not
    static string solve(int[] arr, int n)
    {
        // Initialize flag as true
        bool flag = true;

        // Make unit digit of every element of array with
        // either 0 or 2
        for (int i = 0; i < n; i++) {
            while (arr[i] % 10 != 0 && arr[i] % 10 != 2) {
                arr[i] += arr[i] % 10;
            }
        }

        // Check if it is possible to make the array
        // elements identical or not
        for (int i = 0; i < n; i++) {
            // Elements for 0 as unit digit
            if ((arr[0] % 10) == 0 && (arr[i] != arr[0])) {
                flag = false;
                break;
            }

            // Elements for 2 as unit digit
            if ((arr[i] - arr[0]) % 20 != 0) {
                flag = false;
                break;
            }
        }

        if (flag)
            return "Possible";
        else
            return "Impossible";
    }

    static public void Main()
    {

        // Code
        int[] arr = { 1, 2, 4, 8, 24 };
        int N = arr.Length;

        // Function call
        Console.Write(solve(arr, N));
    }
}

// This code is contributed by lokeshmvs21.

<script>
    // JavaScript code for the approach

// Function to check whether the elements
// of array can be equal or not
function solve(arr, n)
{
    // Initialize flag as 1
    let flag = true;
 
    // Make unit digit of every element of array
    // with either 0 or 2
    for (let i = 0; i < n; i++) {
        while (arr[i] % 10 != 0 && arr[i] % 10 != 2)
            arr[i] += arr[i] % 10;
    }
 
    // Check if it is possible to make
    // the array elements identical or not
    for (let i = 0; i < n; i++) {
 
        // Elements for 0 as unit digit
        if ((arr[0] % 10 == 0) && (arr[i] != arr[0])) {
            flag = false;
            break;
        }
 
        // Elements for 2 as unit digit
        if (((arr[i] - arr[0]) % 20) != 0) {
            flag = false;
            break;
        }
    }
 
    if (flag )
        return "Possible";
    else
        return "Impossible";
}

    // Driver code

    // Given input
    let arr = [ 1, 2, 4, 8, 24 ];
    let N = arr.length;
 
    // Function call
    document.write( solve(arr, N));

// This code is contributed by code_hunt.
</script>

Possible

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