[section:gcd_lcm Greatest Common Divisor and Least Common Multiple]

[section Introduction]

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.

[endsect]

[section Synopsis]

   namespace boost
   {
   namespace integer
   {

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

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

   template <typename I>
   std::pair<typename std::iterator_traits<I>::value_type, I>
      gcd_range(I first, I last);
   template <typename I>
   std::pair<typename std::iterator_traits<I>::value_type, I>
      lcm_range(I first, I last);

   typedef ``['see-below]`` static_gcd_type;

   template < static_gcd_type Value1, static_gcd_type Value2 >
      struct static_gcd;
   template < static_gcd_type Value1, static_gcd_type Value2 >
      struct static_lcm;

   }
   }

[endsect]

[section GCD Function Object]

[*Header: ] [@../../../../boost/integer/common_factor_rt.hpp <boost/integer/common_factor_rt.hpp>]

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

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

The boost::integer::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.

Note that these function objects are `constexpr` in C++14 and later only.
They are also declared `noexcept` when appropriate.

[endsect]

[section LCM Function Object]

[*Header: ] [@../../../../boost/integer/common_factor_rt.hpp <boost/integer/common_factor_rt.hpp>]

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

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

The boost::integer::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.

Note that these function objects are constexpr in C++14 and later only.
They are also declared `noexcept` when appropriate.

[endsect]

[section:run_time Run-time GCD & LCM Determination]

[*Header: ] [@../../../../boost/integer/common_factor_rt.hpp <boost/integer/common_factor_rt.hpp>]

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

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

   template < typename IntegerType, typename... Args >
      constexpr IntegerType  gcd( IntegerType const &a, IntegerType const &b, Args const&... );

   template < typename IntegerType, typename... Args >
      constexpr IntegerType  lcm( IntegerType const &a, IntegerType const &b, Args const&... );

   template <typename I>
   std::pair<typename std::iterator_traits<I>::value_type, I>
      gcd_range(I first, I last);

   template <typename I>
   std::pair<typename std::iterator_traits<I>::value_type, I>
      lcm_range(I first, I last);

The boost::integer::gcd function template returns the greatest common
(nonnegative) divisor of the two integers passed to it.
`boost::integer::gcd_range` is the iteration of the above gcd algorithm over a
range, returning the greatest common divisor of all the elements. The algorithm
terminates when the gcd reaches unity or the end of the range. Thus it also
returns the iterator after the last element inspected because this may not be
equal to the end of the range.  The variadic version of `gcd` behaves similarly
but does not indicate which input value caused the gcd to reach unity.

The boost::integer::lcm function template returns the least common
(nonnegative) multiple of the two integers passed to it.
As with gcd, there are range and variadic versions of the function for
more than 2 arguments.

Note that these functions are constexpr in C++14 and later only.
They are also declared `noexcept` when appropriate.

[endsect]

[section:compile_time Compile time GCD and LCM determination]

[note These functions are deprecated in favor of constexpr `gcd` and `lcm` on C++14 capable compilers.]

[*Header: ] [@../../../../boost/integer/common_factor_ct.hpp <boost/integer/common_factor_ct.hpp>]

   typedef ``['unspecified]`` static_gcd_type;

   template < static_gcd_type Value1, static_gcd_type Value2 >
   struct boost::integer::static_gcd : public mpl::integral_c<static_gcd_type, implementation_defined>
   {
   };

   template < static_gcd_type Value1, static_gcd_type Value2 >
   struct boost::integer::static_lcm : public mpl::integral_c<static_gcd_type, implementation_defined>
   {
   };

The type `static_gcd_type` is the widest unsigned-integer-type that is supported
for use in integral-constant-expressions by the compiler.  Usually this
the same type as `boost::uintmax_t`, but may fall back to being `unsigned long`
for some older compilers.

The boost::integer::static_gcd and boost::integer::static_lcm class templates
take two value-based template parameters of the ['static_gcd_type] type
and inherit from the type `boost::mpl::integral_c`.
Inherited from the base class, they have a member /value/
that 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 `static_gcd_type`.

[h3 Example]

   #include <boost/integer/common_factor.hpp>
   #include <algorithm>
   #include <iterator>
   #include <iostream>

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

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

      cout << "The GCD and LCM of 8 and 9 are "
      << boost::integer::static_gcd<8, 9>::value
      << " and "
      << boost::integer::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::integer::gcd_evaluator<int>() );
      std::copy( c, c + 3, std::ostream_iterator<int>(cout, " ") );
   }

[endsect]

[section:gcd_header Header <boost/integer/common_factor.hpp>]

This header simply includes the headers
[@../../../../boost/integer/common_factor_ct.hpp <boost/integer/common_factor_ct.hpp>]
and [@../../../../boost/integer/common_factor_rt.hpp <boost/integer/common_factor_rt.hpp>].

Note this is a legacy header: it used to contain the actual implementation,
but the compile-time and run-time facilities
were moved to separate headers (since they were independent of each other).

[endsect]

[section:demo Demonstration Program]

The program [@../../../../libs/integer/test/common_factor_test.cpp 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.)

[endsect]

[section 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 maintenance.

[endsect]

[section:gcd_history History]

* 24th April 2017 Moved to Jeremy Murphy's improved algorithms, added constexpr and noexcept support,
added compiler intrinsic support, added variadic and range based versions of the algorithms.
* 13 May 2013 Moved into main Boost.Math Quickbook documentation.
* 17 Dec 2005: Converted documentation to Quickbook Format.
* 2 Jul 2002: Compile-time and run-time items separated to new headers.
* 7 Nov 2001: Initial version

[endsect]

[section:gcd_credits 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.

[endsect]

[endsect]

[/
Copyright 2005, 2013 Daryle Walker.
Distributed under the Boost Software License, Version 1.0.
(See accompanying file LICENSE_1_0.txt or copy at
https://www.boost.org/LICENSE_1_0.txt).
]