# Greatest Common Divisor
Least Common Multiple

The class and function templates in <boost/math/common_factor.hpp> provide run-time and compile-time evaluation of the greatest common divisor (GCD) or least common multiple (LCM) of two integers. These facilities are useful for many numeric-oriented generic programming problems.

## Contents

- Contents
- Synopsis
- GCD Function Object
- LCM Function Object
- Run-time GCD & LCM Determination
- Compile-time GCD & LCM Determination
- Example
- Demonstration Program
- Rationale
- Credits

## Synopsis

>     namespace boost
>     {
>     namespace math
>     {

>     template < typename IntegerType >
>         class gcd_evaluator;
>     template < typename IntegerType >
>         class lcm_evaluator;

>     template < typename IntegerType >
>         IntegerType  gcd( IntegerType const &a, IntegerType const &b );
>     template < typename IntegerType >
>         IntegerType  lcm( IntegerType const &a, IntegerType const &b );

>     template < unsigned long Value1, unsigned long Value2 >
>         struct static_gcd;
>     template < unsigned long Value1, unsigned long Value2 >
>         struct static_lcm;

>     }
>     }

## GCD Function Object

>     template < typename IntegerType >
>     class boost::math::gcd_evaluator
>     {
>     public:
>         // Types
>         typedef IntegerType  result_type;
>         typedef IntegerType  first_argument_type;
>         typedef IntegerType  second_argument_type;

>         // Function object interface
>         result_type  operator ()( first_argument_type const &a,
>          second_argument_type const &b ) const;
>     };

The `boost::math::gcd_evaluator` class template defines a function object class to return the greatest common divisor of two integers. The template is parameterized by a single type, called `IntegerType` here. This type should be a numeric type that represents integers. The result of the function object is always nonnegative, even if either of the operator arguments is negative.

This function object class template is used in the corresponding version of the GCD function template. If a numeric type wants to customize evaluations of its greatest common divisors, then the type should specialize on the `gcd_evaluator` class template.

## LCM Function Object

>     template < typename IntegerType >
>     class boost::math::lcm_evaluator
>     {
>     public:
>         // Types
>         typedef IntegerType  result_type;
>         typedef IntegerType  first_argument_type;
>         typedef IntegerType  second_argument_type;

>         // Function object interface
>         result_type  operator ()( first_argument_type const &a,
>          second_argument_type const &b ) const;
>     };

The `boost::math::lcm_evaluator` class template defines a function object class to return the least common multiple of two integers. The template is parameterized by a single type, called `IntegerType` here. This type should be a numeric type that represents integers. The result of the function object is always nonnegative, even if either of the operator arguments is negative. If the least common multiple is beyond the range of the integer type, the results are undefined.

This function object class template is used in the corresponding version of the LCM function template. If a numeric type wants to customize evaluations of its least common multiples, then the type should specialize on the `lcm_evaluator` class template.

## Run-time GCD & LCM Determination

>     template < typename IntegerType >
>     IntegerType  boost::math::gcd( IntegerType const &a, IntegerType const &b );

>     template < typename IntegerType >
>     IntegerType  boost::math::lcm( IntegerType const &a, IntegerType const &b );

The `boost::math::gcd` function template returns the greatest common (nonnegative) divisor of the two integers passed to it. The `boost::math::lcm` function template returns the least common (nonnegative) multiple of the two integers passed to it. The function templates are parameterized on the function arguments' IntegerType, which is also the return type. Internally, these function templates use an object of the corresponding version of the `gcd_evaluator` and `lcm_evaluator` class templates, respectively.

## Compile-time GCD & LCM Determination

>     template < unsigned long Value1, unsigned long Value2 >
>     struct boost::math::static_gcd
>     {
>         static unsigned long const  value = implementation_defined;
>     };

>     template < unsigned long Value1, unsigned long Value2 >
>     struct boost::math::static_lcm
>     {
>         static unsigned long const  value = implementation_defined;
>     };

The `boost::math::static_gcd` and `boost::math::static_lcm` class templates take two value-based template parameters of the `unsigned long` type and have a single static constant data member, `value`, of that same type. The value of that member is the greatest common factor or least common multiple, respectively, of the template arguments. A compile-time error will occur if the least common multiple is beyond the range of an `unsigned long`.

## Example

>     #include <boost/math/common_factor.hpp>
>     #include <algorithm>
>     #include <iterator>

>     int main()
>     {
>         using std::cout;
>         using std::endl;

>         cout << "The GCD and LCM of 6 and 15 are "
>          << boost::math::gcd(6, 15) << " and "
>          << boost::math::lcm(6, 15) << ", respectively."
>          << endl;

>         cout << "The GCD and LCM of 8 and 9 are "
>          << boost::math::static_gcd<8, 9>::value
>          << " and "
>          << boost::math::static_lcm<8, 9>::value
>          << ", respectively." << endl;

>         int  a[] = { 4, 5, 6 }, b[] = { 7, 8, 9 }, c[3];
>         std::transform( a, a + 3, b, c, boost::math::gcd_evaluator<int>() );
>         std::copy( c, c + 3, std::ostream_iterator<int>(cout, " ") );
>     }

## Demonstration Program

The program common_factor_test.cpp is a demonstration of the results from instantiating various examples of the run-time GCD and LCM function templates and the compile-time GCD and LCM class templates. (The run-time GCD and LCM class templates are tested indirectly through the run-time function templates.)

## Rationale

The greatest common divisor and least common multiple functions are greatly used in some numeric contexts, including some of the other Boost libraries. Centralizing these functions to one header improves code factoring and eases maintainence.

## Credits

The author of the Boost compilation of GCD and LCM computations is Daryle Walker. The code was prompted by existing code hiding in the implementations of Paul Moore's rational library and Steve Cleary's pool library. The code had updates by Helmut Zeisel.

Revised November 7, 2001

© Copyright Daryle Walker 2001. Permission to copy, use, modify, sell and distribute this document is granted provided this copyright notice appears in all copies. This document is provided "as is" without express or implied warranty, and with no claim as to its suitability for any purpose.