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gcd?

File: /projects/sage/sage-7.3/local/lib/python2.7/site-packages/sage/arith/misc.py
Signature : gcd(a, b=None, **kwds)
Docstring :
The greatest common divisor of a and b, or if a is a list and b is
omitted the greatest common divisor of all elements of a.

INPUT:

* "a,b" - two elements of a ring with gcd or

* "a" - a list or tuple of elements of a ring with gcd

Additional keyword arguments are passed to the respectively called
methods.

OUTPUT:

The given elements are first coerced into a common parent. Then,
their greatest common divisor *in that common parent* is returned.

EXAMPLES:

   sage: GCD(97,100)
   1
   sage: GCD(97*10^15, 19^20*97^2)
   97
   sage: GCD(2/3, 4/5)
   2/15
   sage: GCD([2,4,6,8])
   2
   sage: GCD(srange(0,10000,10))  # fast  !!
   10

Note that to take the gcd of n elements for n not= 2 you must put
the elements into a list by enclosing them in "[..]".  Before #4988
the following wrongly returned 3 since the third parameter was just
ignored:

   sage: gcd(3,6,2)
   Traceback (most recent call last):
   ...
   TypeError: gcd() takes at most 2 arguments (3 given)
   sage: gcd([3,6,2])
   1

Similarly, giving just one element (which is not a list) gives an
error:

   sage: gcd(3)
   Traceback (most recent call last):
   ...
   TypeError: 'sage.rings.integer.Integer' object is not iterable

By convention, the gcd of the empty list is (the integer) 0:

   sage: gcd([])
   0
   sage: type(gcd([]))
   <type 'sage.rings.integer.Integer'>

help(euler_phi)

Help on instance of Euler_Phi in module sage.arith.misc:

class Euler_Phi
 |  Return the value of the Euler phi function on the integer n. We
 |  defined this to be the number of positive integers <= n that are
 |  relatively prime to n. Thus if n<=0 then
 |  ``euler_phi(n)`` is defined and equals 0.
 |  
 |  INPUT:
 |  
 |  
 |  -  ``n`` - an integer
 |  
 |  
 |  EXAMPLES::
 |  
 |      sage: euler_phi(1)
 |      1
 |      sage: euler_phi(2)
 |      1
 |      sage: euler_phi(3)
 |      2
 |      sage: euler_phi(12)
 |      4
 |      sage: euler_phi(37)
 |      36
 |  
 |  Notice that euler_phi is defined to be 0 on negative numbers and
 |  0.
 |  
 |  ::
 |  
 |      sage: euler_phi(-1)
 |      0
 |      sage: euler_phi(0)
 |      0
 |      sage: type(euler_phi(0))
 |      <type 'sage.rings.integer.Integer'>
 |  
 |  We verify directly that the phi function is correct for 21.
 |  
 |  ::
 |  
 |      sage: euler_phi(21)
 |      12
 |      sage: [i for i in range(21) if gcd(21,i) == 1]
 |      [1, 2, 4, 5, 8, 10, 11, 13, 16, 17, 19, 20]
 |  
 |  The length of the list of integers 'i' in range(n) such that the
 |  gcd(i,n) == 1 equals euler_phi(n).
 |  
 |  ::
 |  
 |      sage: len([i for i in range(21) if gcd(21,i) == 1]) == euler_phi(21)
 |      True
 |  
 |  The phi function also has a special plotting method.
 |  
 |  ::
 |  
 |      sage: P = plot(euler_phi, -3, 71)
 |  
 |  AUTHORS:
 |  
 |  - William Stein
 |  
 |  - Alex Clemesha (2006-01-10): some examples
 |  
 |  Methods defined here:
 |  
 |  __call__(self, n)
 |      Calls the euler_phi function.
 |      
 |      EXAMPLES::
 |      
 |          sage: Euler_Phi()(10)
 |          4
 |          sage: Euler_Phi()(720)
 |          192
 |  
 |  __repr__(self)
 |      Returns a string describing this class.
 |      
 |      EXAMPLES::
 |      
 |          sage: Euler_Phi().__repr__()
 |          'Number of positive integers <=n but relatively prime to n'
 |  
 |  plot(self, xmin=1, xmax=50, pointsize=30, rgbcolor=(0, 0, 1), join=True, **kwds)
 |      Plot the Euler phi function.
 |      
 |      INPUT:
 |      
 |      
 |      -  ``xmin`` - default: 1
 |      
 |      -  ``xmax`` - default: 50
 |      
 |      -  ``pointsize`` - default: 30
 |      
 |      -  ``rgbcolor`` - default: (0,0,1)
 |      
 |      -  ``join`` - default: True; whether to join the
 |         points.
 |      
 |      -  ``**kwds`` - passed on
 |      
 |      EXAMPLES::
 |      
 |          sage: p = Euler_Phi().plot()
 |          sage: p.ymax()
 |          46.0

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