'''simple docstring'''


import math


def    _lowerCamelCase       ( lowerCamelCase_   :				int  ):



     """simple docstring"""





     if n <= 0:
          raise ValueError('Only positive integers have prime factors'  )
     UpperCAmelCase_				:  str  = []
     while n % 2 == 0:
          pf.append(2  )
          UpperCAmelCase_				:  Tuple  = int(n / 2  )
     for i in range(3      ,	int(math.sqrt(lowerCamelCase_  )  ) + 1      ,	2  ):
          while n % i == 0:
               pf.append(lowerCamelCase_  )
               UpperCAmelCase_				:  Tuple  = int(n / i  )
     if n > 2:
          pf.append(lowerCamelCase_  )
     return pf


def    _lowerCamelCase       ( lowerCamelCase_   :				int  ):



     """simple docstring"""





     if n <= 0:
          raise ValueError('Only positive numbers are accepted'  )
     UpperCAmelCase_				:  List[Any]  = 1
     UpperCAmelCase_				:  List[str]  = 1
     while n % 2 == 0:
          temp += 1
          UpperCAmelCase_				:  Any  = int(n / 2  )
     div *= temp
     for i in range(3      ,	int(math.sqrt(lowerCamelCase_  )  ) + 1      ,	2  ):
          UpperCAmelCase_				:  Any  = 1
          while n % i == 0:
               temp += 1
               UpperCAmelCase_				:  List[Any]  = int(n / i  )
          div *= temp
     if n > 1:
          div *= 2
     return div


def    _lowerCamelCase       ( lowerCamelCase_   :				int  ):



     """simple docstring"""





     if n <= 0:
          raise ValueError('Only positive numbers are accepted'  )
     UpperCAmelCase_				:  Any  = 1
     UpperCAmelCase_				:  Union[str, Any]  = 1
     while n % 2 == 0:
          temp += 1
          UpperCAmelCase_				:  str  = int(n / 2  )
     if temp > 1:
          s *= (2**temp - 1) / (2 - 1)
     for i in range(3      ,	int(math.sqrt(lowerCamelCase_  )  ) + 1      ,	2  ):
          UpperCAmelCase_				:  Dict  = 1
          while n % i == 0:
               temp += 1
               UpperCAmelCase_				:  str  = int(n / i  )
          if temp > 1:
               s *= (i**temp - 1) / (i - 1)
     return int(lowerCamelCase_  )


def    _lowerCamelCase       ( lowerCamelCase_   :				int  ):



     """simple docstring"""





     UpperCAmelCase_				:  List[str]  = n
     for x in set(prime_factors(lowerCamelCase_  )  ):
          s *= (x - 1) / x
     return int(lowerCamelCase_  )


if __name__ == "__main__":
 import doctest

 doctest.testmod()











from __future__ import annotations

class        _A			:



       def __init__(						self				:	str	,						__SCREAMING_SNAKE_CASE				:	int = 0):


           '''simple docstring'''

           __a					=							value
           __a					=							None
           __a					=							None





def 	__snake_case      (     _UpperCAmelCase  ,					_UpperCAmelCase							):
    if treea is None:
        return treea
    if treea is None:
        return treea

    __a					=							treea.value + treea.value
    __a					=							merge_two_binary_trees(treea.left  ,					treea.left							)
    __a					=							merge_two_binary_trees(treea.right  ,					treea.right							)
    return treea





def 	__snake_case      (     _UpperCAmelCase							):
    if root:
        print(root.value							)
        print_preorder(root.left							)
        print_preorder(root.right							)


if __name__ == "__main__":
 __snake_case		:List[Any] 						=	Node(1)
 __snake_case		:Optional[Any] 						=	Node(2)
 __snake_case		:Dict 						=	Node(3)
 __snake_case		:int 						=	Node(4)

 __snake_case		:List[str] 						=	Node(2)
 __snake_case		:List[str] 						=	Node(4)
 __snake_case		:Tuple 						=	Node(6)
 __snake_case		:str 						=	Node(9)
 __snake_case		:List[str] 						=	Node(5)

 print('''Tree1 is: ''')
 print_preorder(treea)
 print('''Tree2 is: ''')
 print_preorder(treea)
 __snake_case		:Union[str, Any] 						=	merge_two_binary_trees(treea, treea)
 print('''Merged Tree is: ''')
 print_preorder(merged_tree)





