# Nth number in a set of multiples of A , B or C

Given four integers **N**, **A**, **B** and **C**. The task is to print the **N ^{th}** number in the set containing the multiples of

**A**,

**B**or

**C**.

**Examples:**

Input:A = 2, B = 3, C = 5, N = 8

Output:10

2, 3, 4, 5, 6, 8, 9,10, 12, 14, …

Input:A = 2, B = 3, C = 5, N = 100

Output:136

**Naive approach:** Start traversing from **1** until we find the **N ^{th}** element which is divisible by either

**A**,

**B**or

**C**.

**Efficient approach:** Given a number we can find the count of the divisors of either **A**, **B** or **C**. Now, binary search can be used to find the **N ^{th}** number which is divisible by either

**A**,

**B**or

**C**.

So, if the number is num then

**count = (num/A) + (num/B) + (num/C) – (num/lcm(A, B)) – (num/lcm(C, B)) – (num/lcm(A, C)) – (num/lcm(A, B, C))**

Below is the implementation of the above approach:

## C++

`// C++ program to find nth term ` `// divisible by a, b or c ` ` ` `#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 return the count of integers ` `// from the range [1, num] which are ` `// divisible by either a, b or c ` `long` `divTermCount(` `long` `a, ` `long` `b, ` `long` `c, ` `long` `num) ` `{ ` ` ` `// Calculate the number of terms divisible by a, b ` ` ` `// and c then remove the terms which are divisible ` ` ` `// by both (a, b) or (b, c) or (c, a) and then ` ` ` `// add the numbers which are divisible by a, b and c ` ` ` `return` `((num / a) + (num / b) + (num / c) ` ` ` `- (num / ((a * b) / gcd(a, b))) ` ` ` `- (num / ((c * b) / gcd(c, b))) ` ` ` `- (num / ((a * c) / gcd(a, c))) ` ` ` `+ (num / ((a * b * c) / gcd(gcd(a, b), c)))); ` `} ` ` ` `// Function for binary search to find the ` `// nth term divisible by a, b or c ` `int` `findNthTerm(` `int` `a, ` `int` `b, ` `int` `c, ` `long` `n) ` `{ ` ` ` `// Set low to 1 and high to LONG_MAX ` ` ` `long` `low = 1, high = LONG_MAX, mid; ` ` ` ` ` `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, c, 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() ` `{ ` ` ` `long` `a = 2, b = 3, c = 5, n = 100; ` ` ` ` ` `cout << findNthTerm(a, b, c, n); ` ` ` ` ` `return` `0; ` `} ` |

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## Java

`// Java program to find nth term ` `// divisible by a, b or c ` `class` `GFG ` `{ ` ` ` ` ` `// Function to return ` ` ` `// gcd of a and b ` ` ` `static` `long` `gcd(` `long` `a, ` `long` `b) ` ` ` `{ ` ` ` `if` `(a == ` `0` `) ` ` ` `{ ` ` ` `return` `b; ` ` ` `} ` ` ` `return` `gcd(b % a, a); ` ` ` `} ` ` ` ` ` `// Function to return the count of integers ` ` ` `// from the range [1, num] which are ` ` ` `// divisible by either a, b or c ` ` ` `static` `long` `divTermCount(` `long` `a, ` `long` `b, ` ` ` `long` `c, ` `long` `num) ` ` ` `{ ` ` ` `// Calculate the number of terms divisible by a, b ` ` ` `// and c then remove the terms which are divisible ` ` ` `// by both (a, b) or (b, c) or (c, a) and then ` ` ` `// add the numbers which are divisible by a, b and c ` ` ` `return` `((num / a) + (num / b) + (num / c) - ` ` ` `(num / ((a * b) / gcd(a, b))) - ` ` ` `(num / ((c * b) / gcd(c, b))) - ` ` ` `(num / ((a * c) / gcd(a, c))) + ` ` ` `(num / ((a * b * c) / gcd(gcd(a, b), c)))); ` ` ` `} ` ` ` ` ` `// Function for binary search to find the ` ` ` `// nth term divisible by a, b or c ` ` ` `static` `long` `findNthTerm(` `int` `a, ` `int` `b, ` `int` `c, ` `long` `n) ` ` ` `{ ` ` ` ` ` `// Set low to 1 and high to LONG_MAX ` ` ` `long` `low = ` `1` `, high = Long.MAX_VALUE, mid; ` ` ` ` ` `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, c, 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 = ` `3` `, c = ` `5` `, n = ` `100` `; ` ` ` ` ` `System.out.println(findNthTerm(a, b, c, n)); ` ` ` `} ` `} ` ` ` `// This code is contributed by 29AjayKumar ` |

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## Python3

`# Python3 program to find nth term ` `# divisible by a, b or c ` `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 return the count of integers ` `# from the range [1, num] which are ` `# divisible by either a, b or c ` `def` `divTermCount(a, b, c, num): ` ` ` ` ` `# Calculate the number of terms divisible by a, b ` ` ` `# and c then remove the terms which are divisible ` ` ` `# by both (a, b) or (b, c) or (c, a) and then ` ` ` `# add the numbers which are divisible by a, b and c ` ` ` `return` `((num ` `/` `a) ` `+` `(num ` `/` `b) ` `+` `(num ` `/` `c) ` `-` ` ` `(num ` `/` `((a ` `*` `b) ` `/` `gcd(a, b))) ` `-` ` ` `(num ` `/` `((c ` `*` `b) ` `/` `gcd(c, b))) ` `-` ` ` `(num ` `/` `((a ` `*` `c) ` `/` `gcd(a, c))) ` `+` ` ` `(num ` `/` `((a ` `*` `b ` `*` `c) ` `/` `gcd(gcd(a, b), c)))); ` ` ` `# Function for binary search to find the ` `# nth term divisible by a, b or c ` `def` `findNthTerm(a, b, c, n): ` ` ` ` ` `# Set low to 1 and high to LONG_MAX ` ` ` `low ` `=` `1` `; high ` `=` `sys.maxsize; mid ` `=` `0` `; ` ` ` ` ` `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, c, mid) < n): ` ` ` `low ` `=` `mid ` `+` `1` `; ` ` ` ` ` `# If current term is greater than equal to ` ` ` `# n then high = mid ` ` ` `else` `: ` ` ` `high ` `=` `mid; ` ` ` ` ` `return` `int` `(low); ` ` ` `# Driver code ` `a ` `=` `2` `; b ` `=` `3` `; c ` `=` `5` `; n ` `=` `100` `; ` ` ` `print` `(findNthTerm(a, b, c, n)); ` ` ` `# This code is contributed by 29AjayKumar ` |

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## C#

`// C# program to find nth term ` `// divisible by a, b or c ` `using` `System; ` ` ` `class` `GFG ` `{ ` ` ` ` ` `// Function to return ` ` ` `// gcd of a and b ` ` ` `static` `long` `gcd(` `long` `a, ` `long` `b) ` ` ` `{ ` ` ` `if` `(a == 0) ` ` ` `{ ` ` ` `return` `b; ` ` ` `} ` ` ` `return` `gcd(b % a, a); ` ` ` `} ` ` ` ` ` `// Function to return the count of integers ` ` ` `// from the range [1, num] which are ` ` ` `// divisible by either a, b or c ` ` ` `static` `long` `divTermCount(` `long` `a, ` `long` `b, ` ` ` `long` `c, ` `long` `num) ` ` ` `{ ` ` ` `// Calculate the number of terms divisible by a, b ` ` ` `// and c then remove the terms which are divisible ` ` ` `// by both (a, b) or (b, c) or (c, a) and then ` ` ` `// add the numbers which are divisible by a, b and c ` ` ` `return` `((num / a) + (num / b) + (num / c) - ` ` ` `(num / ((a * b) / gcd(a, b))) - ` ` ` `(num / ((c * b) / gcd(c, b))) - ` ` ` `(num / ((a * c) / gcd(a, c))) + ` ` ` `(num / ((a * b * c) / gcd(gcd(a, b), c)))); ` ` ` `} ` ` ` ` ` `// Function for binary search to find the ` ` ` `// nth term divisible by a, b or c ` ` ` `static` `long` `findNthTerm(` `int` `a, ` `int` `b, ` ` ` `int` `c, ` `long` `n) ` ` ` `{ ` ` ` ` ` `// Set low to 1 and high to LONG_MAX ` ` ` `long` `low = 1, high = ` `long` `.MaxValue, mid; ` ` ` ` ` `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, c, 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 = 3, c = 5, n = 100; ` ` ` ` ` `Console.WriteLine(findNthTerm(a, b, c, n)); ` ` ` `} ` `} ` ` ` `// This code is contributed by PrinciRaj1992 ` |

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**Output:**

136

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