# Least Common Denominator (LCD)

The **lowest Common Denominator** or **Least Common Denominator** is the Least Common Multiple of the denominators of a set of fractions.

**Common denominator** : when the denominators of two or more fractions are the same.

Least Common denominator is the smallest of all common denominators.

Why do we need **LCD** ?

It simplifies addition, subtraction and comparing fraction.

Common Denominator can be simply evaluated by multiplying the denominators. In this case, 3 * 6 = 18

But that may not always be least common denominator, as in this case LCD = 6 and not 18. LCD is actually LCM of denominators.

**Examples : **

LCD for fractions 5/12 and 7/15 is 60. We can write both fractions as 25/60 and 28/60 so that they can be added and subtracted easily. LCD for fractions 1/3 and 4/7 is 21.

Example Problem : Given two fractions, find their sum using least common dominator.

**Examples**:

Input : 1/6 + 7/15 Output : 19/30 Explanation : LCM of 6 and 15 is 30. So, 5/30 + 14/30 = 19/30 Input : 1/3 + 1/6 Output : 3/6 Explanation : LCM of 3 and 6 is 6. So, 2/6 + 1/6 = 3/6

**Note*** These answers can be further simplified by Anomalous cancellation.

## C++

`// C++ Program to determine ` `// LCD of two fractions and ` `// Perform addition on fractions ` `#include <iostream> ` `using` `namespace` `std; ` ` ` `// function to calculate gcd ` `// or hcf of two numbers. ` `int` `gcd(` `int` `a, ` `int` `b) ` `{ ` ` ` `if` `(a == 0) ` ` ` `return` `b; ` ` ` `return` `gcd(b % a, a); ` `} ` ` ` `// function to calculate ` `// lcm of two numbers. ` `int` `lcm(` `int` `a, ` `int` `b) ` `{ ` ` ` `return` `(a * b) / gcd(a, b); ` `} ` ` ` `void` `printSum(` `int` `num1, ` `int` `den1, ` ` ` `int` `num2, ` `int` `den2) ` `{ ` ` ` `// least common multiple ` ` ` `// of denominators LCD ` ` ` `// of 6 and 15 is 30. ` ` ` `int` `lcd = lcm(den1, den2); ` ` ` ` ` `// Computing the numerators for LCD: ` ` ` `// Writing 1/6 as 5/30 and 7/15 as ` ` ` `// 14/30 ` ` ` `num1 *= (lcd / den1); ` ` ` `num2 *= (lcd / den2); ` ` ` ` ` `// Our sum is going to be res_num/lcd ` ` ` `int` `res_num = num1 + num2; ` ` ` `cout << res_num << ` `"/"` `<< lcd; ` `} ` ` ` `// Driver Code ` `int` `main() ` `{ ` ` ` `// First fraction is 1/6 ` ` ` `int` `num1 = 1, den1 = 6; ` ` ` ` ` `// Second fraction is 7/15 ` ` ` `int` `num2 = 7, den2 = 15; ` ` ` ` ` `printSum(num1, den1, num2, den2); ` ` ` `return` `0; ` `} ` |

*chevron_right*

*filter_none*

## Java

`// Java Program to determine LCD of two ` `// fractions and Perform addition on ` `// fractions ` `public` `class` `GFG { ` ` ` ` ` `// function to calculate gcd or ` ` ` `// hcf of two numbers. ` ` ` `static` `int` `gcd(` `int` `a, ` `int` `b) ` ` ` `{ ` ` ` `if` `(a == ` `0` `) ` ` ` `return` `b; ` ` ` ` ` `return` `gcd(b % a, a); ` ` ` `} ` ` ` ` ` `// function to calculate lcm of ` ` ` `// two numbers. ` ` ` `static` `int` `lcm(` `int` `a, ` `int` `b) ` ` ` `{ ` ` ` `return` `(a * b) / gcd(a, b); ` ` ` `} ` ` ` ` ` `static` `void` `printSum(` `int` `num1, ` `int` `den1, ` ` ` `int` `num2, ` `int` `den2) ` ` ` `{ ` ` ` ` ` `// least common multiple of ` ` ` `// denominators LCD of 6 and 15 ` ` ` `// is 30. ` ` ` `int` `lcd = lcm(den1, den2); ` ` ` ` ` `// Computing the numerators for LCD: ` ` ` `// Writing 1/6 as 5/30 and 7/15 as ` ` ` `// 14/30 ` ` ` `num1 *= (lcd / den1); ` ` ` `num2 *= (lcd / den2); ` ` ` ` ` `// Our sum is going to be res_num/lcd ` ` ` `int` `res_num = num1 + num2; ` ` ` ` ` `System.out.print( res_num + ` `"/"` `+ lcd); ` ` ` `} ` ` ` ` ` `// Driver code ` ` ` `public` `static` `void` `main(String args[]) ` ` ` `{ ` ` ` ` ` `// First fraction is 1/6 ` ` ` `int` `num1 = ` `1` `, den1 = ` `6` `; ` ` ` ` ` `// Second fraction is 7/15 ` ` ` `int` `num2 = ` `7` `, den2 = ` `15` `; ` ` ` ` ` `printSum(num1, den1, num2, den2); ` ` ` `} ` `} ` ` ` `// This code is contributed by Sam007. ` |

*chevron_right*

*filter_none*

## Python3

`# python Program to determine ` `# LCD of two fractions and ` `# Perform addition on fractions ` ` ` `# function to calculate gcd ` `# or hcf of two numbers. ` `def` `gcd(a, b): ` ` ` ` ` `if` `(a ` `=` `=` `0` `): ` ` ` `return` `b ` ` ` `return` `gcd(b ` `%` `a, a) ` ` ` ` ` `# function to calculate ` `# lcm of two numbers. ` `def` `lcm(a, b): ` ` ` ` ` `return` `(a ` `*` `b) ` `/` `gcd(a, b) ` ` ` ` ` `def` `printSum(num1, den1, ` ` ` `num2, den2): ` ` ` ` ` `# least common multiple ` ` ` `# of denominators LCD ` ` ` `# of 6 and 15 is 30. ` ` ` `lcd ` `=` `lcm(den1, den2); ` ` ` ` ` `# Computing the numerators ` ` ` `# for LCD: Writing 1/6 as ` ` ` `# 5/30 and 7/15 as 14/30 ` ` ` `num1 ` `*` `=` `(lcd ` `/` `den1) ` ` ` `num2 ` `*` `=` `(lcd ` `/` `den2) ` ` ` ` ` `# Our sum is going to be ` ` ` `# res_num/lcd ` ` ` `res_num ` `=` `num1 ` `+` `num2; ` ` ` `print` `( ` `int` `(res_num) , ` `"/"` `, ` ` ` `int` `(lcd)) ` ` ` `# Driver Code ` `# First fraction is 1/6 ` `num1 ` `=` `1` `den1 ` `=` `6` ` ` `# Second fraction is 7/15 ` `num2 ` `=` `7` `den2 ` `=` `15` `printSum(num1, den1, num2, den2); ` ` ` `# This code is contributed ` `# by Sam007 ` |

*chevron_right*

*filter_none*

## C#

`// C# Program to determine LCD of two ` `// fractions and Perform addition on ` `// fractions ` `using` `System; ` ` ` `class` `GFG { ` ` ` ` ` `// function to calculate gcd or ` ` ` `// hcf of two numbers. ` ` ` `static` `int` `gcd(` `int` `a, ` `int` `b) ` ` ` `{ ` ` ` `if` `(a == 0) ` ` ` `return` `b; ` ` ` ` ` `return` `gcd(b % a, a); ` ` ` `} ` ` ` ` ` `// function to calculate lcm of ` ` ` `// two numbers. ` ` ` `static` `int` `lcm(` `int` `a, ` `int` `b) ` ` ` `{ ` ` ` `return` `(a * b) / gcd(a, b); ` ` ` `} ` ` ` ` ` `static` `void` `printSum(` `int` `num1, ` `int` `den1, ` ` ` `int` `num2, ` `int` `den2) ` ` ` `{ ` ` ` ` ` `// least common multiple of ` ` ` `// denominators LCD of 6 and 15 ` ` ` `// is 30. ` ` ` `int` `lcd = lcm(den1, den2); ` ` ` ` ` `// Computing the numerators for LCD: ` ` ` `// Writing 1/6 as 5/30 and 7/15 as ` ` ` `// 14/30 ` ` ` `num1 *= (lcd / den1); ` ` ` `num2 *= (lcd / den2); ` ` ` ` ` `// Our sum is going to be res_num/lcd ` ` ` `int` `res_num = num1 + num2; ` ` ` ` ` `Console.Write( res_num + ` `"/"` `+ lcd); ` ` ` `} ` ` ` ` ` `// Driver code ` ` ` `public` `static` `void` `Main () ` ` ` `{ ` ` ` ` ` `// First fraction is 1/6 ` ` ` `int` `num1 = 1, den1 = 6; ` ` ` ` ` `// Second fraction is 7/15 ` ` ` `int` `num2 = 7, den2 = 15; ` ` ` ` ` `printSum(num1, den1, num2, den2); ` ` ` `} ` `} ` ` ` `// This code is contributed by Sam007. ` |

*chevron_right*

*filter_none*

## PHP

`<?php ` `// PHP Program to determine ` `// LCD of two fractions and ` `// Perform addition on fractions ` ` ` `// function to calculate gcd ` `// or hcf of two numbers. ` `function` `gcd(` `$a` `,` `$b` `) ` `{ ` ` ` `if` `(` `$a` `== 0) ` ` ` `return` `$b` `; ` ` ` `return` `gcd(` `$b` `% ` `$a` `, ` `$a` `); ` `} ` ` ` `// function to calculate ` `// lcm of two numbers. ` `function` `lcm(` `$a` `,` `$b` `) ` `{ ` ` ` `return` `(` `$a` `* ` `$b` `) / gcd(` `$a` `, ` `$b` `); ` `} ` ` ` `function` `printSum(` `$num1` `, ` `$den1` `, ` ` ` `$num2` `, ` `$den2` `) ` `{ ` ` ` `// least common multiple ` ` ` `// of denominators ` ` ` `// LCD of 6 and 15 is 30. ` ` ` `$lcd` `= lcm(` `$den1` `, ` `$den2` `); ` ` ` ` ` `// Computing the numerators for LCD: ` ` ` `// Writing 1/6 as 5/30 and 7/15 as ` ` ` `// 14/30 ` ` ` `$num1` `*= (` `$lcd` `/ ` `$den1` `); ` ` ` `$num2` `*= (` `$lcd` `/ ` `$den2` `); ` ` ` ` ` `// Our sum is going to be res_num/lcd ` ` ` `$res_num` `= ` `$num1` `+ ` `$num2` `; ` ` ` `echo` `$res_num` `. ` `"/"` `. ` `$lcd` `; ` `} ` ` ` `// Driver Code ` ` ` `// First fraction is 1/6 ` ` ` `$num1` `= 1; ` ` ` `$den1` `= 6; ` ` ` ` ` `// Second fraction is 7/15 ` ` ` `$num2` `= 7; ` ` ` `$den2` `= 15; ` ` ` ` ` `printSum(` `$num1` `, ` `$den1` `, ` `$num2` `, ` `$den2` `); ` ` ` `// This code is contributed by Sam007. ` `?> ` |

*chevron_right*

*filter_none*

**Output :**

19/30

This article is contributed by **Shubham Rana**. If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.

## Recommended Posts:

- Largest proper fraction with sum of numerator and denominator equal to a given number
- Find ΔX which is added to numerator and denominator both of fraction (a/b) to convert it to another fraction (c/d)
- Number of triangles formed by joining vertices of n-sided polygon with two common sides and no common sides
- Sum of common divisors of two numbers A and B
- Common Divisors of Two Numbers
- Common prime factors of two numbers
- Print the kth common factor of two numbers
- Count divisors of n that have at-least one digit common with n
- Divide the two given numbers by their common divisors
- C++ Program for Common Divisors of Two Numbers
- Maximum count of common divisors of A and B such that all are co-primes to one another
- Count of common multiples of two numbers in a range
- Count common prime factors of two numbers
- Nearest element with at-least one common prime factor
- Longest arithmetic progression with the given common difference