Count triplets such that product of two numbers added with third number is N
Last Updated :
11 Oct, 2021
Given a positive integer N, the task is to find the number of triplets (A, B, C) where A, B, C are positive integers such that the product of two numbers added with the third number is N i.e., A * B + C = N.
Examples:
Input: N = 3
Output: 3
Explanation:
Following are the possible triplets satisfying the given criteria:
- (1, 1, 2): The value of 1*1 + 2 = 3.
- (1, 2, 1): The value of 1*2 + 1 = 3.
- (2, 1, 1): The value of 2*1 + 1 = 3.
Therefore, the total count of such triplets is 3.
Input: N = 5
Output: 8
Approach: The given problem can be solved by rearranging the equation A * B + C = N as A * B = N - C. Now, the only possible values A and B can have to satisfy the above equation is the divisors of N - C. For Example, if the value of N - C = 18, having 6 divisors that are 1, 2, 3, 6, 9, 18. So, values of A, B satisfying the above equation are: (1, 18), (2, 9), (3, 6), (6, 3), (9, 2), (18, 1). So, for the value of N - C = 18, possible values of A, B are 6, i.e., the number of divisors of N - C(= 18). Follow the steps below to solve the given problem:
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 divisors of
// the number (N - i)
int countDivisors(int n)
{
// Stores the resultant count of
// divisors of (N - i)
int divisors = 0;
int i;
// Iterate over range [1, sqrt(N)]
for (i = 1; i * i < n; i++) {
if (n % i == 0) {
divisors++;
}
}
if (i - (n / i) == 1) {
i--;
}
for (; i >= 1; i--) {
if (n % i == 0) {
divisors++;
}
}
// Return the total divisors
return divisors;
}
// Function to find the number of triplets
// such that A * B - C = N
int possibleTriplets(int N)
{
int count = 0;
// Loop to fix the value of C
for (int i = 1; i < N; i++) {
// Adding the number of
// divisors in count
count += countDivisors(N - i);
}
// Return count of triplets
return count;
}
// Driver Code
int main()
{
int N = 10;
cout << possibleTriplets(N);
return 0;
}
Java
// Java program for the above approach
import java.io.*;
class GFG
{
// Function to find the divisors of
// the number (N - i)
static int countDivisors(int n)
{
// Stores the resultant count of
// divisors of (N - i)
int divisors = 0;
int i;
// Iterate over range [1, sqrt(N)]
for (i = 1; i * i < n; i++) {
if (n % i == 0) {
divisors++;
}
}
if (i - (n / i) == 1) {
i--;
}
for (; i >= 1; i--) {
if (n % i == 0) {
divisors++;
}
}
// Return the total divisors
return divisors;
}
// Function to find the number of triplets
// such that A * B - C = N
static int possibleTriplets(int N)
{
int count = 0;
// Loop to fix the value of C
for (int i = 1; i < N; i++) {
// Adding the number of
// divisors in count
count += countDivisors(N - i);
}
// Return count of triplets
return count;
}
// Driver Code
public static void main (String[] args) {
int N = 10;
System.out.println(possibleTriplets(N));
}
}
// This code is contributed by Dharanendra L V.
Python3
# Python program for the above approach
import math
# function to find the divisors of
# the number (N - i)
def countDivisors(n):
# Stores the resultant count of
# divisors of (N - i)
divisors = 0
# Iterate over range [1, sqrt(N)]
for i in range(1, math.ceil(math.sqrt(n))+1):
if n % i == 0:
divisors = divisors+1
if (i - (n / i) == 1):
i = i-1
for i in range(math.ceil(math.sqrt(n))+1, 1, -1):
if (n % i == 0):
divisors = divisors+1
# Return the total divisors
return divisors
# def to find the number of triplets
# such that A * B - C = N
def possibleTriplets(N):
count = 0
# Loop to fix the value of C
for i in range(1, N):
# Adding the number of
# divisors in count
count = count + countDivisors(N - i)
# Return count of triplets
return count
# Driver Code
# Driver Code
if __name__ == "__main__":
N = 10
print(possibleTriplets(N))
# This code is contributed by Potta Lokesh
C#
// C# program for the above approach
using System;
using System.Collections.Generic;
class GFG{
// Function to find the divisors of
// the number (N - i)
static int countDivisors(int n)
{
// Stores the resultant count of
// divisors of (N - i)
int divisors = 0;
int i;
// Iterate over range [1, sqrt(N)]
for (i = 1; i * i < n; i++) {
if (n % i == 0) {
divisors++;
}
}
if (i - (n / i) == 1) {
i--;
}
for (; i >= 1; i--) {
if (n % i == 0) {
divisors++;
}
}
// Return the total divisors
return divisors;
}
// Function to find the number of triplets
// such that A * B - C = N
static int possibleTriplets(int N)
{
int count = 0;
// Loop to fix the value of C
for (int i = 1; i < N; i++) {
// Adding the number of
// divisors in count
count += countDivisors(N - i);
}
// Return count of triplets
return count;
}
// Driver Code
public static void Main()
{
int N = 10;
Console.Write(possibleTriplets(N));
}
}
// This code is contributed by SURENDRA_GANGWAR.
JavaScript
<script>
// javascript program for the above approach
// Function to find the divisors of
// the number (N - i)
function countDivisors(n)
{
// Stores the resultant count of
// divisors of (N - i)
var divisors = 0;
var i;
// Iterate over range [1, sqrt(N)]
for (i = 1; i * i < n; i++) {
if (n % i == 0) {
divisors++;
}
}
if (i - (n / i) == 1) {
i--;
}
for (; i >= 1; i--) {
if (n % i == 0) {
divisors++;
}
}
// Return the total divisors
return divisors;
}
// Function to find the number of triplets
// such that A * B - C = N
function possibleTriplets(N)
{
var count = 0;
// Loop to fix the value of C
for (var i = 1; i < N; i++) {
// Adding the number of
// divisors in count
count += countDivisors(N - i);
}
// Return count of triplets
return count;
}
// Driver Code
var N = 10;
document.write(possibleTriplets(N));
// This code is contributed by shikhasingrajput
</script>
Time Complexity: O(N*sqrt(N))
Auxiliary Space: O(1)
Similar Reads
DSA Tutorial - Learn Data Structures and Algorithms DSA (Data Structures and Algorithms) is the study of organizing data efficiently using data structures like arrays, stacks, and trees, paired with step-by-step procedures (or algorithms) to solve problems effectively. Data structures manage how data is stored and accessed, while algorithms focus on
7 min read
Quick Sort QuickSort is a sorting algorithm based on the Divide and Conquer that picks an element as a pivot and partitions the given array around the picked pivot by placing the pivot in its correct position in the sorted array. It works on the principle of divide and conquer, breaking down the problem into s
12 min read
Merge Sort - Data Structure and Algorithms Tutorials Merge sort is a popular sorting algorithm known for its efficiency and stability. It follows the divide-and-conquer approach. It works by recursively dividing the input array into two halves, recursively sorting the two halves and finally merging them back together to obtain the sorted array. Merge
14 min read
Data Structures Tutorial Data structures are the fundamental building blocks of computer programming. They define how data is organized, stored, and manipulated within a program. Understanding data structures is very important for developing efficient and effective algorithms. What is Data Structure?A data structure is a st
2 min read
Bubble Sort Algorithm Bubble Sort is the simplest sorting algorithm that works by repeatedly swapping the adjacent elements if they are in the wrong order. This algorithm is not suitable for large data sets as its average and worst-case time complexity are quite high.We sort the array using multiple passes. After the fir
8 min read
Breadth First Search or BFS for a Graph Given a undirected graph represented by an adjacency list adj, where each adj[i] represents the list of vertices connected to vertex i. Perform a Breadth First Search (BFS) traversal starting from vertex 0, visiting vertices from left to right according to the adjacency list, and return a list conta
15+ min read
Binary Search Algorithm - Iterative and Recursive Implementation Binary Search Algorithm is a searching algorithm used in a sorted array by repeatedly dividing the search interval in half. The idea of binary search is to use the information that the array is sorted and reduce the time complexity to O(log N). Binary Search AlgorithmConditions to apply Binary Searc
15 min read
Insertion Sort Algorithm Insertion sort is a simple sorting algorithm that works by iteratively inserting each element of an unsorted list into its correct position in a sorted portion of the list. It is like sorting playing cards in your hands. You split the cards into two groups: the sorted cards and the unsorted cards. T
9 min read
Array Data Structure Guide In this article, we introduce array, implementation in different popular languages, its basic operations and commonly seen problems / interview questions. An array stores items (in case of C/C++ and Java Primitive Arrays) or their references (in case of Python, JS, Java Non-Primitive) at contiguous
4 min read
Sorting Algorithms A Sorting Algorithm is used to rearrange a given array or list of elements in an order. For example, a given array [10, 20, 5, 2] becomes [2, 5, 10, 20] after sorting in increasing order and becomes [20, 10, 5, 2] after sorting in decreasing order. There exist different sorting algorithms for differ
3 min read