GCD-LCM
=======

Save all functions into the same file named ``gcdlcm.py``.

#. Write the funcion :py:func:`gcd` that takes two positive integers
   *a* and *b*, and returns its greatest common divisor. Proposed method:
   using the following sequence *min(a,b)*, *min(a,b)-1, ..., 2, 1*, the
   first element of this sequence that divides both *a* and *b* is its
   greatest common divisor. Examples:
   
   .. literalinclude:: gcd-1.txt
         :language: python3
         :lines: 2-

   .. note::
         More tests are provided in file :download:`gcd-1.txt<gcd-1.txt>`

#. Write the function :py:func:`gcd_euclides_1` that takes two positive
   integers *a* and *b*, and returns its greatest common divisor. Use
   the Euclides method based on the following *gcd* properties:

   - :math:`gcd(a,b)=gcd(b,a)`
   - if :math:`a>b` then :math:`gcd(a,b)=gcd(a-b,b)`
   - if :math:`b>a` then :math:`gcd(a,b)=gcd(a,b-a)`
   - :math:`gcd(a,a)=a`
     
   Examples:
   :math:`gcd(5,15)=gcd(5,10)=gcd(5,5)=5`
	 
      .. literalinclude:: gcd-2.txt
         :language: python3
         :lines: 2-

   .. note::
          More tests are provided in file :download:`gcd-2.txt <gcd-2.txt>`

#. Write the function :py:func:`gcd_euclides_2` that takes two
   positive integers *a* and *b*, and returns its greatest common
   divisor. Use the `Euclides method
   <https://en.wikipedia.org/wiki/Euclidean_algorithm>`__ based on the
   following *gcd* properties:
   
   - if :math:`a=c\times b+r`, then :math:`gcd(a,b)=gcd(b,r)`
   - :math:`gcd(a, 0)= a`

   Examples: :math:`gcd(567,28)=gcd(28,7)=gcd(7,0)=7`
 
      .. literalinclude:: gcd-3.txt
         :language: python3
         :lines: 2-

   .. note::
          More tests are provided in file :download:`gcd-3.txt <gcd-3.txt>`

#. Write the function :py:func:`lcm` that takes two positive
   integers *a* and *b*, and returns its `least common multiple
   <https://en.wikipedia.org/wiki/Least_common_multiple>`__ . Use some
   of the previous functions. Examples:

      .. literalinclude:: lcm.txt
         :language: python3
         :lines: 2-

   .. note::
         More tests are provided in file :download:`lcm.txt <lcm.txt>`

.. rubric:: Solution

A solution of these functions is provided in the :download:`gcdlcm.py <solutions/gcdlcm.py>` file.