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Count of distinct integers belonging to first N terms of at least one of given GPs

  • Difficulty Level : Medium
  • Last Updated : 29 Sep, 2021

Given two Geometric Progressions (a1, r1) and (a2, r2) where (x, y) represents GP with initial term and common ratio y and an integer N, the task is to find the count of the distinct integers that belong to the first N terms of at least one of the given geometric progressions.

Examples:

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Input: N = 5, a1 = 3, r1 = 2, a2 = 6, r2 = 2
Output: 6
Explanation: The first 5 terms of the given geometric progressions are {3, 6, 12, 24, 48} and {6, 12, 24, 48, 96} respectively. Hence, the total count of distinct integers in the GP is 6.

Input: N = 5, a1 = 3, r1 = 2, a2 = 2, r2 = 3
Output: 9
Explanation: The first 5 terms of the given geometric progressions are {3, 6, 12, 24, 48} and {2, 6, 18, 54, 162} respectively. Hence, the total count of distinct integers in the GP is 9.



Approach: The given problem can be solved by the observation that the total count of distinct integers can be calculated by generating the first N terms of both the Geometric Progressions and removing the duplicates terms. This can be achieved by the use of the set data structure. Firstly, generate all the N terms of the 1st GP and insert the terms into a set S. Similarly, insert the terms of the 2nd GP into the set S. The size of the set S is the required 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 find the count of distinct
// integers that belong to the first N
// terms of at least one of them is GP
int UniqueGeometricTerms(int N, int a1,
                         int r1, int a2,
                         int r2)
{
    // Stores the integers that occur in
    // GPs in a set data-structure
    set<int> S;
 
    // Stores the current integer of
    // the first GP
    long long p1 = a1;
 
    // Iterate first N terms of first GP
    for (int i = 0; i < N; i++) {
 
        // Insert the ith term of GP in S
        S.insert(p1);
        p1 = (long long)(p1 * r1);
    }
 
    // Stores the current integer
    // of the second GP
    long long p2 = a2;
 
    // Iterate first N terms of second GP
    for (int i = 0; i < N; i++) {
        S.insert(p2);
        p2 = (long long)(p2 * r2);
    }
 
    // Return Answer
    return S.size();
}
 
// Driver Code
int main()
{
    int N = 5;
    int a1 = 3, r1 = 2, a2 = 2, r2 = 3;
 
    cout << UniqueGeometricTerms(
        N, a1, r1, a2, r2);
 
    return 0;
}

Java




// Java program for the above approach
import java.util.*;
 
class GFG
{
 
// Function to find the count of distinct
// integers that belong to the first N
// terms of at least one of them is GP
static int UniqueGeometricTerms(int N, int a1,
                         int r1, int a2,
                         int r2)
{
    // Stores the integers that occur in
    // GPs in a set data-structure
    HashSet<Integer> S=new HashSet<Integer>();
 
    // Stores the current integer of
    // the first GP
    int p1 = a1;
 
    // Iterate first N terms of first GP
    for (int i = 0; i < N; i++) {
 
        // Insert the ith term of GP in S
        S.add(p1);
        p1 = (p1 * r1);
    }
 
    // Stores the current integer
    // of the second GP
    int p2 = a2;
 
    // Iterate first N terms of second GP
    for (int i = 0; i < N; i++) {
        S.add(p2);
        p2 = (p2 * r2);
    }
 
    // Return Answer
    return S.size();
}
 
// Driver Code
public static void main(String[] args)
{
    int N = 5;
    int a1 = 3, r1 = 2, a2 = 2, r2 = 3;
 
    System.out.print(UniqueGeometricTerms(
        N, a1, r1, a2, r2));
 
}
}
 
// This code is contributed by shikhasingrajput

Python3




# Python 3 program for the above approach
 
# Function to find the count of distinct
# integers that belong to the first N
# terms of at least one of them is GP
def UniqueGeometricTerms(N, a1, r1, a2, r2):
   
    # Stores the integers that occur in
    # GPs in a set data-structure
    S = set()
 
    # Stores the current integer of
    # the first GP
    p1 = a1
 
    # Iterate first N terms of first GP
    for i in range(N):
       
        # Insert the ith term of GP in S
        S.add(p1)
        p1 = (p1 * r1)
 
    # Stores the current integer
    # of the second GP
    p2 = a2
 
    # Iterate first N terms of second GP
    for i in range(N):
        S.add(p2)
        p2 = (p2 * r2)
 
    # Return Answer
    return len(S)
 
# Driver Code
if __name__ == '__main__':
    N = 5
    a1 = 3
    r1 = 2
    a2 = 2
    r2 = 3
 
    print(UniqueGeometricTerms(N, a1, r1, a2, r2))
 
    # This code is contributed by SURENDRA_GANGWAR.

C#




// C# program for the above approach
using System;
using System.Collections.Generic;
 
public class GFG
{
 
// Function to find the count of distinct
// integers that belong to the first N
// terms of at least one of them is GP
static int UniqueGeometricTerms(int N, int a1,
                         int r1, int a2,
                         int r2)
{
    // Stores the integers that occur in
    // GPs in a set data-structure
    HashSet<int> S=new HashSet<int>();
 
    // Stores the current integer of
    // the first GP
    int p1 = a1;
 
    // Iterate first N terms of first GP
    for (int i = 0; i < N; i++) {
 
        // Insert the ith term of GP in S
        S.Add(p1);
        p1 = (p1 * r1);
    }
 
    // Stores the current integer
    // of the second GP
    int p2 = a2;
 
    // Iterate first N terms of second GP
    for (int i = 0; i < N; i++) {
        S.Add(p2);
        p2 = (p2 * r2);
    }
 
    // Return Answer
    return S.Count;
}
 
// Driver Code
public static void Main(string[] args)
{
    int N = 5;
    int a1 = 3, r1 = 2, a2 = 2, r2 = 3;
 
    Console.Write(UniqueGeometricTerms(
        N, a1, r1, a2, r2));
}
}
 
// This code is contributed by AnkThon

Javascript




<script>
        // JavaScript Program to implement
        // the above approach
 
        // Function to find the count of distinct
        // integers that belong to the first N
        // terms of at least one of them is GP
        function UniqueGeometricTerms(N, a1,
            r1, a2,
            r2)
       {
        
            // Stores the integers that occur in
            // GPs in a set data-structure
            let S = new Set();
             
            // Stores the current integer of
            // the first GP
            let p1 = a1;
 
            // Iterate first N terms of first GP
            for (let i = 0; i < N; i++) {
 
                // Insert the ith term of GP in S
                S.add(p1);
                p1 = (p1 * r1);
            }
 
            // Stores the current integer
            // of the second GP
            let p2 = a2;
 
            // Iterate first N terms of second GP
            for (let i = 0; i < N; i++) {
                S.add(p2);
                p2 = (p2 * r2);
            }
 
            // Return Answer
            return S.size;
        }
 
        // Driver Code
 
        let N = 5;
        let a1 = 3, r1 = 2, a2 = 2, r2 = 3;
 
        document.write(UniqueGeometricTerms(
            N, a1, r1, a2, r2));
 
     // This code is contributed by Potta Lokesh
 
    </script>
Output: 
9

 

Time Complexity: O(N*log N)
Auxiliary Space: O(N)




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