# Program to find Nth term divisible by a or b

Given two integers and . The task is to find the Nth term which is divisible by either of or .

**Examples :**

Input: a = 2, b = 5, N = 10Output: 16Input: a = 3, b = 7, N = 25Output: 57

**Naive Approach :** A simple approach is to traverse over all the terms starting from 1 until we find the desired Nth term which is divisible by either of or . This solution has time complexity of O(N).

**Efficient Approach :** The idea is to use Binary search. Here we can calculate how many numbers from 1 to are divisible by either a or b by using formula:

All the multiples of lcm(a, b) will be divisible by both and so we need to remove these terms. Now if the number of divisible terms is less than N we will increase the low position of binary search otherwise decrease high until number of divisible terms is equal to N.

Below is the implementation of the above idea :

## C++

`// C++ program to find nth term ` `// divisible by a or b ` ` ` `#include <bits/stdc++.h> ` `using` `namespace` `std; ` ` ` `// Function to return ` `// gcd of a and b ` `int` `gcd(` `int` `a, ` `int` `b) ` `{ ` ` ` `if` `(a == 0) ` ` ` `return` `b; ` ` ` ` ` `return` `gcd(b % a, a); ` `} ` ` ` `// Function to calculate how many numbers ` `// from 1 to num are divisible by a or b ` `int` `divTermCount(` `int` `a, ` `int` `b, ` `int` `lcm, ` `int` `num) ` `{ ` ` ` `// calculate number of terms divisible by a and ` ` ` `// by b then, remove the terms which is are ` ` ` `// divisible by both a and b ` ` ` `return` `num / a + num / b - num / lcm; ` `} ` ` ` `// Binary search to find the nth term ` `// divisible by a or b ` `int` `findNthTerm(` `int` `a, ` `int` `b, ` `int` `n) ` `{ ` ` ` `// set low to 1 and high to max(a, b)*n, here ` ` ` `// we have taken high as 10^18 ` ` ` `int` `low = 1, high = INT_MAX, mid; ` ` ` `int` `lcm = (a * b) / gcd(a, b); ` ` ` ` ` `while` `(low < high) { ` ` ` `mid = low + (high - low) / 2; ` ` ` ` ` `// if the current term is less than ` ` ` `// n then we need to increase low ` ` ` `// to mid + 1 ` ` ` `if` `(divTermCount(a, b, lcm, mid) < n) ` ` ` `low = mid + 1; ` ` ` ` ` `// if current term is greater than equal to ` ` ` `// n then high = mid ` ` ` `else` ` ` `high = mid; ` ` ` `} ` ` ` ` ` `return` `low; ` `} ` ` ` `// Driver code ` `int` `main() ` `{ ` ` ` `int` `a = 2, b = 5, n = 10; ` ` ` `cout << findNthTerm(a, b, n) << endl; ` ` ` ` ` `return` `0; ` `} ` |

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## Java

`// Java program to find nth term ` `// divisible by a or b ` `class` `GFG ` `{ ` `// Function to return ` `// gcd of a and b ` `static` `int` `gcd(` `int` `a, ` `int` `b) ` `{ ` ` ` `if` `(a == ` `0` `) ` ` ` `return` `b; ` ` ` ` ` `return` `gcd(b % a, a); ` `} ` ` ` `// Function to calculate how many numbers ` `// from 1 to num are divisible by a or b ` `static` `int` `divTermCount(` `int` `a, ` `int` `b, ` ` ` `int` `lcm, ` `int` `num) ` `{ ` ` ` `// calculate number of terms ` ` ` `// divisible by a and by b then, ` ` ` `// remove the terms which is are ` ` ` `// divisible by both a and b ` ` ` `return` `num / a + num / b - num / lcm; ` `} ` ` ` `// Binary search to find the ` `// nth term divisible by a or b ` `static` `int` `findNthTerm(` `int` `a, ` `int` `b, ` `int` `n) ` `{ ` ` ` `// set low to 1 and high to max(a, b)*n, ` ` ` `// here we have taken high as 10^18 ` ` ` `int` `low = ` `1` `, high = Integer.MAX_VALUE, mid; ` ` ` `int` `lcm = (a * b) / gcd(a, b); ` ` ` ` ` `while` `(low < high) ` ` ` `{ ` ` ` `mid = low + (high - low) / ` `2` `; ` ` ` ` ` `// if the current term is less ` ` ` `// than n then we need to increase ` ` ` `// low to mid + 1 ` ` ` `if` `(divTermCount(a, b, lcm, mid) < n) ` ` ` `low = mid + ` `1` `; ` ` ` ` ` `// if current term is greater ` ` ` `// than equal to n then high = mid ` ` ` `else` ` ` `high = mid; ` ` ` `} ` ` ` ` ` `return` `low; ` `} ` ` ` `// Driver code ` `public` `static` `void` `main (String[] args) ` `{ ` ` ` `int` `a = ` `2` `, b = ` `5` `, n = ` `10` `; ` ` ` `System.out.println(findNthTerm(a, b, n)); ` `} ` `} ` ` ` `// This code is contributed by Smitha ` |

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## Python3

`# Python 3 program to find nth term ` `# divisible by a or b ` `import` `sys ` ` ` `# Function to return gcd of a and b ` `def` `gcd(a, b): ` ` ` `if` `a ` `=` `=` `0` `: ` ` ` `return` `b ` ` ` `return` `gcd(b ` `%` `a, a) ` ` ` `# Function to calculate how many numbers ` `# from 1 to num are divisible by a or b ` `def` `divTermCount(a, b, lcm, num): ` ` ` ` ` `# calculate number of terms divisible ` ` ` `# by a and by b then, remove the terms ` ` ` `# which are divisible by both a and b ` ` ` `return` `num ` `/` `/` `a ` `+` `num ` `/` `/` `b ` `-` `num ` `/` `/` `lcm ` ` ` `# Binary search to find the nth term ` `# divisible by a or b ` `def` `findNthTerm(a, b, n): ` ` ` ` ` `# set low to 1 and high to max(a, b)*n, ` ` ` `# here we have taken high as 10^18 ` ` ` `low ` `=` `1` `; high ` `=` `sys.maxsize ` ` ` `lcm ` `=` `(a ` `*` `b) ` `/` `/` `gcd(a, b) ` ` ` `while` `low < high: ` ` ` `mid ` `=` `low ` `+` `(high ` `-` `low) ` `/` `/` `2` ` ` ` ` `# if the current term is less ` ` ` `# than n then we need to increase ` ` ` `# low to mid + 1 ` ` ` `if` `divTermCount(a, b, lcm, mid) < n: ` ` ` `low ` `=` `mid ` `+` `1` ` ` ` ` `# if current term is greater ` ` ` `# than equal to n then high = mid ` ` ` `else` `: ` ` ` `high ` `=` `mid ` ` ` `return` `low ` ` ` `# Driver code ` `a ` `=` `2` `; b ` `=` `5` `; n ` `=` `10` `print` `(findNthTerm(a, b, n)) ` ` ` `# This code is contributed by Shrikant13 ` |

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## C#

`// C# program to find nth term ` `// divisible by a or b ` `using` `System; ` ` ` `class` `GFG ` `{ ` `// Function to return gcd of a and b ` `static` `int` `gcd(` `int` `a, ` `int` `b) ` `{ ` ` ` `if` `(a == 0) ` ` ` `return` `b; ` ` ` ` ` `return` `gcd(b % a, a); ` `} ` ` ` `// Function to calculate how many numbers ` `// from 1 to num are divisible by a or b ` `static` `int` `divTermCount(` `int` `a, ` `int` `b, ` ` ` `int` `lcm, ` `int` `num) ` `{ ` ` ` `// calculate number of terms ` ` ` `// divisible by a and by b then, ` ` ` `// remove the terms which is are ` ` ` `// divisible by both a and b ` ` ` `return` `num / a + num / b - num / lcm; ` `} ` ` ` `// Binary search to find the ` `// nth term divisible by a or b ` `static` `int` `findNthTerm(` `int` `a, ` `int` `b, ` `int` `n) ` `{ ` ` ` `// set low to 1 and high to max(a, b)*n, ` ` ` `// here we have taken high as 10^18 ` ` ` `int` `low = 1, high = ` `int` `.MaxValue, mid; ` ` ` `int` `lcm = (a * b) / gcd(a, b); ` ` ` ` ` `while` `(low < high) ` ` ` `{ ` ` ` `mid = low + (high - low) / 2; ` ` ` ` ` `// if the current term is less ` ` ` `// than n then we need to increase ` ` ` `// low to mid + 1 ` ` ` `if` `(divTermCount(a, b, lcm, mid) < n) ` ` ` `low = mid + 1; ` ` ` ` ` `// if current term is greater ` ` ` `// than equal to n then high = mid ` ` ` `else` ` ` `high = mid; ` ` ` `} ` ` ` ` ` `return` `low; ` `} ` ` ` `// Driver code ` `static` `public` `void` `Main () ` `{ ` ` ` `int` `a = 2, b = 5, n = 10; ` ` ` `Console.WriteLine(findNthTerm(a, b, n)); ` `} ` `} ` ` ` `// This code is contributed by Sach_Code ` |

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**Output:**

16

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