# 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; ` `} ` |

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## 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. ` |

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## 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 ` |

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## 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. ` |

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## 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. ` `?> ` |

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**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.

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