Number of co-prime pairs from 1 to N which consists of given two digits

Given an integer N and two integers D1 and D2 ( < 10), the task is to find the number of co-prime pairs less than or equal to N consisting of only digits D1 and D2.

Examples:

Input: N = 30, D1 = 2, D2 = 3
Output: 5
Explanation: 
All possible pairs of numbers upto 30 having digits 2 and 3 which are co-prime to each other are (2, 3), (2, 23), (3, 22), (3, 23), (22, 23) .

Input: N = 100, D1 = 5, D2 = 6
Output: 8
Explanation: 
All possible pairs of numbers upto 100 having digits 5 and 6 which are co-prime to each other are (5, 6), (5, 56), (5, 66), (6, 55), (6, 65), (55, 56), (56, 65), (65, 66).

Approach: The idea to solve this problem is based on the following observations:

Observation:

• Find and append every number consisting only given two digits which are less than or equal to N into a list.
• Now it’s easy to find the number of unordered pairs which are co-primes.
• Note that there can be maximum 1 + 2 + 2² + 2³ + 2⁴ + ………2¹⁰= 2047 numbers in the list i.e. Overall Time Complexity cannot exceed O(2047 * 2047), as the maximum no. of digits possible is 9.

Follow the steps below to solve the problem:

• Initialize an empty list, say l, and append the given two digits as a string into the list.
• Sort the list.
• Initialize two lists, say total and temp2 for further use.
• Iterate until l[0] < 10:
  • Append the given two digits as a string to all the elements present in the list l.
  • Keep updating l in the way shown below:
    • [2, 3] -> [‘2’ + ‘2’, ‘2’ + ‘3’, ‘3’ + ‘2’, ‘3’ + ‘3’]
    • [22, 23, 32, 33] – > [’22’ + ‘2’, ’22’ + ‘3’, ’23’ + ‘2’, ’23’ + ‘3’, ’32’ + ‘2’, ’32’ + ‘3’, ’33’ + ‘2’, ’33’ + ‘3’] and so on.
• Define a function numOfPairs() which returns the count of unordered co-prime pairs.
• Print the count returned by the above function as the answer.

Below is the implementation of the above approach:

C++

// C++ program for the above approach

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

// Function to check whether given
// integers are co-prime or not
int coprime(int a, int b) { return (__gcd(a, b) == 1); }

// Utility function to count
// number of co-prime pairs
int numOfPairs(vector<string> arr, int N)
{
    int count = 0;

    // Traverse the array
    for (int i = 0; i < N - 1; i++) {
        for (int j = i + 1; j < N; j++) {

            // If co-prime
            if (coprime(stoi(arr[i]), stoi(arr[j]))) {

                // Increment count
                count = count + 1;
            }
        }
    }
    // Return count
    return count;
}

// Function to count number
// of co-prime pairs
int noOfCoPrimePairs(int N, int d1, int d2)
{

    // Stores digits in string form
    vector<string> l;
    l.push_back(to_string(d1));
    l.push_back(to_string(d2));

    // Sort the list
    sort(l.begin(), l.end());

    if (N < stoi(l[1]))
        return 0;

    // Keep two copies of list l
    vector<string> total = l;
    vector<string> temp2 = l;
    int flag = 0;
    vector<string> temp3;

    // Generate 2 digit numbers
    // using d1 and d2
    while (l[0].length() < 10) {
        for (int i = 0; i < l.size(); i++) {
            for (int j = 0; j < 2; j++) {

                // If current number
                // does not exceed N
                if (stoi(l[i] + temp2[j]) > N) {
                    flag = 1;
                    break;
                }
                total.push_back(l[i] + temp2[j]);
                temp3.push_back(l[i] + temp2[j]);
            }
            if (flag == 1)
                break;
        }
        if (flag == 1)
            break;
        l = temp3;
        vector<string> temp3;
    }

    // Stores length of list
    int lenOfTotal = total.size();

    // Stores number of co-prime pairs
    int ans = numOfPairs(total, lenOfTotal);

    // Print number of co-prime pairs
    cout << (ans);
}

// Driver Code
int main()
{

    // Given value of N, d1, d2
    int N = 30, d1 = 2, d2 = 3;

    // Function call to count
    // number of co-prime pairs
    noOfCoPrimePairs(N, d1, d2);
}

// This code is contributed by ukasp.

Java

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

class GFG{

static int GCD(int a, int b)
{
    return b == 0 ? a : GCD(b, a % b);
}

// Function to check whether given
// integers are co-prime or not
static boolean coprime(int a, int b)
{
    if (GCD(a, b) == 1)
        return true;

    return false;
}

// Utility function to count
// number of co-prime pairs
static int numOfPairs(ArrayList<String> arr, int N)
{
    int count = 0;

    // Traverse the array
    for(int i = 0; i < N - 1; i++)
    {
        for(int j = i + 1; j < N; j++)
        {

            // If co-prime
            if (coprime(Integer.parseInt(arr.get(i)),
                        Integer.parseInt(arr.get(j))))
            {

                // Increment count
                count = count + 1;
            }
        }
    }

    // Return count
    return count;
}

// Function to count number
// of co-prime pairs
static void noOfCoPrimePairs(int N, int d1, int d2)
{

    // Stores digits in string form
    ArrayList<String> l = new ArrayList<String>();
    l.add(Integer.toString(d1));
    l.add(Integer.toString(d2));

    // Sort the list
    Collections.sort(l);

    if (N < Integer.parseInt(l.get(1)))
        return;

    // Keep two copies of list l
    ArrayList<String> total = new ArrayList<String>(l);
    ArrayList<String> temp2 = new ArrayList<String>(l);
    int flag = 0;
    ArrayList<String> temp3 = new ArrayList<String>(l);

    // Generate 2 digit numbers
    // using d1 and d2
    while (l.get(0).length() < 10)
    {
        for(int i = 0; i < l.size(); i++)
        {
            for(int j = 0; j < 2; j++)
            {

                // If current number
                // does not exceed N
                if (Integer.parseInt(l.get(i) +
                                 temp2.get(j)) > N)
                {
                    flag = 1;
                    break;
                }
                total.add(l.get(i) + temp2.get(j));
                temp3.add(l.get(i) + temp2.get(j));
            }
            if (flag == 1)
                break;
        }
        if (flag == 1)
            break;

        l = temp3;
        temp3.clear();
    }

    // Stores length of list
    int lenOfTotal = total.size();

    // Stores number of co-prime pairs
    int ans = numOfPairs(total, lenOfTotal);

    // Print number of co-prime pairs
    System.out.print(ans);
}

// Driver Code
public static void main(String args[])
{

    // Given value of N, d1, d2
    int N = 30, d1 = 2, d2 = 3;

    // Function call to count
    // number of co-prime pairs
    noOfCoPrimePairs(N, d1, d2);
}
}

// This code is contributed by bgangwar59

Python3

# Python3 program for the above approach

from copy import deepcopy
import math

# Function to check whether given
# integers are co-prime or not
def coprime(a, b):
    return (math.gcd(a, b)) == 1

# Utility function to count
# number of co-prime pairs
def numOfPairs(arr, N):
    count = 0

    # Traverse the array
    for i in range(0, N-1):
        for j in range(i+1, N):

            # If co-prime
            if (coprime(int(arr[i]), int(arr[j]))):

                # Increment count
                count = count + 1

    # Return count
    return count

# Function to count number
# of co-prime pairs
def noOfCoPrimePairs(N, d1, d2):

    # Stores digits in string form
    l = []
    l.append(str(d1))
    l.append(str(d2))

    # Sort the list
    l.sort()

    if int(N) < int(l[1]):
        return 0

    # Keep two copies of list l
    total = temp2 = deepcopy(l)
    flag = 0
    temp3 = []

    # Generate 2 digit numbers
    # using d1 and d2
    while len(l[0]) < 10:
        for i in range(len(l)):
            for j in range(2):

                # If current number
                # does not exceed N
                if int(l[i]+temp2[j]) > int(N):
                    flag = 1
                    break

                total.append(l[i]+temp2[j])
                temp3.append(l[i]+temp2[j])

            if flag == 1:
                break
        if flag == 1:
            break
        l = deepcopy(temp3)
        temp3 = []

    # Stores length of list
    lenOfTotal = len(total)

    # Stores number of co-prime pairs
    ans = numOfPairs(total, lenOfTotal)

    # Print number of co-prime pairs
    print(ans)


# Driver Code
if __name__ == "__main__":

    # Given value of N, d1, d2
    N = 30
    d1 = 2
    d2 = 3

    # Function call to count
    # number of co-prime pairs
    noOfCoPrimePairs(N, d1, d2)

C#

// C# program for the above approach
using System;
using System.Collections.Generic;

class GFG{

static int GCD(int a, int b)
{
    return b == 0 ? a : GCD(b, a % b);
}

// Function to check whether given
// integers are co-prime or not
static bool coprime(int a, int b)
{
    if (GCD(a, b) == 1)
        return true;

    return false;
}

// Utility function to count
// number of co-prime pairs
static int numOfPairs(List<string> arr, int N)
{
    int count = 0;

    // Traverse the array
    for(int i = 0; i < N - 1; i++)
    {
        for(int j = i + 1; j < N; j++)
        {

            // If co-prime
            if (coprime(Int32.Parse(arr[i]),
                        Int32.Parse(arr[j])))
            {

                // Increment count
                count = count + 1;
            }
        }
    }

    // Return count
    return count;
}

// Function to count number
// of co-prime pairs
static void noOfCoPrimePairs(int N, int d1, int d2)
{

    // Stores digits in string form
    List<string> l = new List<string>();
    l.Add(d1.ToString());
    l.Add(d2.ToString());

    // Sort the list
    l.Sort();

    if (N < Int32.Parse(l[1]))
        return;

    // Keep two copies of list l
    List<string> total = new List<string>(l);
    List<string> temp2 = new List<string>(l);
    int flag = 0;
    List<string> temp3 = new List<string>();

    // Generate 2 digit numbers
    // using d1 and d2
    while (l[0].Length < 10)
    {
        for(int i = 0; i < l.Count; i++)
        {
            for(int j = 0; j < 2; j++)
            {

                // If current number
                // does not exceed N
                if (Int32.Parse(l[i] + temp2[j]) > N)
                {
                    flag = 1;
                    break;
                }
                total.Add(l[i] + temp2[j]);
                temp3.Add(l[i] + temp2[j]);
            }
            if (flag == 1)
                break;
        }
        if (flag == 1)
            break;

        l = temp3;
        temp3.Clear();
    }

    // Stores length of list
    int lenOfTotal = total.Count;

    // Stores number of co-prime pairs
    int ans = numOfPairs(total, lenOfTotal);

    // Print number of co-prime pairs
    Console.WriteLine(ans);
}

// Driver Code
public static void Main()
{

    // Given value of N, d1, d2
    int N = 30, d1 = 2, d2 = 3;

    // Function call to count
    // number of co-prime pairs
    noOfCoPrimePairs(N, d1, d2);
}
}

// This code is contributed by ipg2016107

Javascript

<script>

// JavaScript program for the above approach

function GCD(a,b)
{
    return b == 0 ? a : GCD(b, a % b);
}

// Function to check whether given
// integers are co-prime or not
function coprime(a,b)
{
    if (GCD(a, b) == 1)
        return true;

    return false;
}

// Utility function to count
// number of co-prime pairs
function numOfPairs(arr,N)
{
    let count = 0;

    // Traverse the array
    for(let i = 0; i < N - 1; i++)
    {
        for(let j = i + 1; j < N; j++)
        {

            // If co-prime
            if (coprime(parseInt(arr[i]),
                        parseInt(arr[j])))
            {

                // Increment count
                count = count + 1;
            }
        }
    }

    // Return count
    return count;
}

// Function to count number
// of co-prime pairs
function noOfCoPrimePairs(N,d1,d2)
{
    // Stores digits in string form
    let l = [];
    l.push(d1.toString());
    l.push(d2.toString());

    // Sort the list
    l.sort();

    if (N < parseInt(l[1]))
        return;

    // Keep two copies of list l
    let total = [...l];
    let temp2 = [...l];
    let flag = 0;
    let temp3 = [];

    // Generate 2 digit numbers
    // using d1 and d2
    while (l[0].length < 10)
    {
        for(let i = 0; i < l.length; i++)
        {
            for(let j = 0; j < 2; j++)
            {

                // If current number
                // does not exceed N
                if (parseInt(l[i] +
                                 temp2[j]) > N)
                {
                    flag = 1;
                    break;
                }
                total.push(l[i] + temp2[j]);
                temp3.push(l[i] + temp2[j]);
            }
            if (flag == 1)
                break;
        }
        if (flag == 1)
            break;

        l = [...temp3];
        temp3=[];
    }

    // Stores length of list
    let lenOfTotal = total.length;

    // Stores number of co-prime pairs
    let ans = numOfPairs(total, lenOfTotal);

    // Print number of co-prime pairs
    document.write(ans);
}

// Driver Code
// Given value of N, d1, d2
let N = 30, d1 = 2, d2 = 3;

// Function call to count
// number of co-prime pairs
noOfCoPrimePairs(N, d1, d2);


// This code is contributed by avanitrachhadiya2155

</script>

Output:

5

Time Complexity: O(2^logN)
Auxiliary Space: O(2^logN)