C Library Functions - least_common_multiple (3)

NAME

least_common_multiple(3f) - [M_factor] Least common multiple of two integers or vector m(:), matrix m(:,:) or cuboid m(:,:,:) (LICENSE:MIT)

Synopsis
Description
Examples
Method
Author
License

SYNOPSIS

use, intrinsic :: iso_fortran_env, only : int64

    integer function least_common_multiple(i,j)
    integer,intent(in):: i,j
     or
    integer function(kind=int32) least_common_multiple(m)
    integer,intent(in):: m(:)
     or
    integer,intent(in):: m(:,:)
     or
    integer,intent(in):: m(:,:,:)

DESCRIPTION

From Wikipedia, the free encyclopedia:
In arithmetic and number theory, the least common multiple (also called the lowest common multiple or smallest common multiple) of two integers a and b, usually denoted by LCM(a, b), is the smallest positive integer that is divisible by both a and b. Since division of integers by zero is undefined, this definition has meaning only if a and b are both different from zero. However, some authors define lcm(a,0) as 0 for all a, which is the result of taking the LCM to be the least upper bound in the lattice of divisibility.
The LCM is familiar from grade-school arithmetic as the "lowest common denominator" (LCD) that must be determined before fractions can be added, subtracted or compared. The LCM of more than two integers is also well-defined: it is the smallest positive integer that is divisible by each of them.

EXAMPLES

Sample Program:

   program demo_least_common_multiple
   use M_factor, only : lcm=>least_common_multiple
   implicit none
      write(*,*)’SCALAR:’
         call writeit(10,24,120)
         call writeit(15,30,30)
         call writeit(-15,-30,30)
         call writeit(15,-30,30)
         call writeit(-15,30,30)

      write(*,*)’VECTOR:’
         call writeit_v([10,24],120)
         call writeit_v([15,30],30)
         call writeit_v([-15,-30],30)
         call writeit_v([5,-15,-40],120)
         call writeit_v([2,3,4,5],60)
      write(*,*)’Special cases:’
         call writeit_v([15,0],0)
         call writeit_v([-15,0],0)
         call writeit_v([0],0)
         call writeit_v([-10],10)
         call writeit_v([22],22)
         call writeit_v([0,0],0)
         call writeit_v([0,0,0,0,0],0)
         call writeit_v([0,0,0,-1,0],0)
         call writeit_v([0,0,0,33,0,3,11],0)
      contains

      subroutine writeit(ii,jj,answer)
      integer,intent(in) :: ii,jj
      integer,intent(in) :: answer
         write(*,’("  For lcm(",I0,",",I0,") the value is ",I0," which is ",L1)’)&
            & ii,jj,lcm(ii,jj),lcm(ii,jj).eq.answer
      end subroutine writeit

      subroutine writeit_v(array,answer)
      integer,intent(in) :: array(:)
      integer,intent(in) :: answer
         write(*,’("  For lcm([",*(i0:,1x))’,advance=’no’)array
         write(*,’("]) the value is ",i0," which is ",L1)’) &
            & lcm(array),lcm(array).eq.answer
      end subroutine writeit_v

   end program demo_least_common_multiple

Expected results:

    > SCALAR:
    >  For lcm(10,24) the value is 120 which is T
    >  For lcm(15,30) the value is 30 which is T
    >  For lcm(-15,-30) the value is 30 which is T
    >  For lcm(15,-30) the value is 30 which is T
    >  For lcm(-15,30) the value is 30 which is T
    > VECTOR:
    >  For lcm([10 24]) the value is 120 which is T
    >  For lcm([15 30]) the value is 30 which is T
    >  For lcm([-15 -30]) the value is 30 which is T
    >  For lcm([5 -15 -40]) the value is 120 which is T
    >  For lcm([2 3 4 5]) the value is 60 which is T
    > Special cases:
    >  For lcm([15 0]) the value is 0 which is T
    >  For lcm([-15 0]) the value is 0 which is T
    >  For lcm([0]) the value is 0 which is T
    >  For lcm([-10]) the value is 10 which is T
    >  For lcm([22]) the value is 22 which is T
    >  For lcm([0 0]) the value is 0 which is T
    >  For lcm([0 0 0 0 0]) the value is 0 which is T
    >  For lcm([0 0 0 -1 0]) the value is 0 which is T
    >  For lcm([0 0 0 33 0 3 11]) the value is 0 which is T

METHOD

Reduction by the greatest common divisor
The following formula reduces the problem of computing the least common multiple to the problem of computing the greatest common divisor (GCD), also known as the greatest common factor:
    lcm(a,b) = |a*b| / gcd(a,b)
This formula is also valid when exactly one of a and b is 0, since gcd(a, 0) = |a|. (However, if both a and b are 0, this formula would cause division by zero; lcm(0, 0) = 0 is a special case.

AUTHOR

John S. Urban

LICENSE

MIT License

Nemo Release 3.1

least_common_multiple (3)

February 23, 2025

Generated by manServer 1.08 from a12adffe-87b8-420f-83ff-262da4a3b29b using man macros.