

'''simple docstring'''

from __future__ import annotations




def  UpperCamelCase(	UpperCAmelCase_    ,			UpperCAmelCase_    ,			UpperCAmelCase_      ):
 assert n > 1 and a > 0 and greatest_common_divisor(UpperCAmelCase_    ,			UpperCAmelCase_      ) == 1
 ((UpperCAmelCase)   ,					(UpperCAmelCase)   ,					(UpperCAmelCase))			:      Dict		   =					extended_gcd(UpperCAmelCase_    ,			UpperCAmelCase_      )  # Implemented below
 UpperCAmelCase			:      int		   =					(b * s) % n
 return x




def  UpperCamelCase(	UpperCAmelCase_    ,			UpperCAmelCase_      ):
 ((UpperCAmelCase)   ,					(UpperCAmelCase))			:      int		   =					extended_euclid(UpperCAmelCase_    ,			UpperCAmelCase_      )  # Implemented below
 if b < 0:
  UpperCAmelCase			:      Optional[int]		   =					(b % n + n) % n
 return b




def  UpperCamelCase(	UpperCAmelCase_    ,			UpperCAmelCase_    ,			UpperCAmelCase_      ):
 UpperCAmelCase			:      Tuple		   =					invert_modulo(UpperCAmelCase_    ,			UpperCAmelCase_      )
 UpperCAmelCase			:      List[Any]		   =					(b * s) % n
 return x




def  UpperCamelCase(	UpperCAmelCase_    ,			UpperCAmelCase_      ):
 assert a >= 0 and b >= 0

 if b == 0:
  UpperCAmelCase   ,					UpperCAmelCase   ,					UpperCAmelCase			:      Dict		   =					a, 1, 0
 else:
  ((UpperCAmelCase)   ,					(UpperCAmelCase)   ,					(UpperCAmelCase))			:      Optional[int]		   =					extended_gcd(UpperCAmelCase_    ,			a % b      )
  UpperCAmelCase			:      Dict		   =					q
  UpperCAmelCase			:      int		   =					p - q * (a // b)

 assert a % d == 0 and b % d == 0
 assert d == a * x + b * y

 return (d, x, y)




def  UpperCamelCase(	UpperCAmelCase_    ,			UpperCAmelCase_      ):
 if b == 0:
  return (1, 0)
 ((UpperCAmelCase)   ,					(UpperCAmelCase))			:      Tuple		   =					extended_euclid(UpperCAmelCase_    ,			a % b      )
 UpperCAmelCase			:      Tuple		   =					a // b
 return (y, x - k * y)




def  UpperCamelCase(	UpperCAmelCase_    ,			UpperCAmelCase_      ):
 if a < b:
  UpperCAmelCase   ,					UpperCAmelCase			:      Union[str, Any]		   =					b, a

 while a % b != 0:
  UpperCAmelCase   ,					UpperCAmelCase			:      int		   =					b, a % b

 return b


if __name__ == "__main__":
 from doctest import testmod

 testmod(name="modular_division", verbose=True)
 testmod(name="modular_division2", verbose=True)
 testmod(name="invert_modulo", verbose=True)
 testmod(name="extended_gcd", verbose=True)
 testmod(name="extended_euclid", verbose=True)
 testmod(name="greatest_common_divisor", verbose=True)







'''simple docstring'''

from __future__ import annotations




def  UpperCamelCase(	UpperCAmelCase_    ,			UpperCAmelCase_    ,			UpperCAmelCase_      ):
 assert n > 1 and a > 0 and greatest_common_divisor(UpperCAmelCase_    ,			UpperCAmelCase_      ) == 1
 ((UpperCAmelCase)   ,					(UpperCAmelCase)   ,					(UpperCAmelCase))			:      Dict		   =					extended_gcd(UpperCAmelCase_    ,			UpperCAmelCase_      )  # Implemented below
 UpperCAmelCase			:      int		   =					(b * s) % n
 return x




def  UpperCamelCase(	UpperCAmelCase_    ,			UpperCAmelCase_      ):
 ((UpperCAmelCase)   ,					(UpperCAmelCase))			:      int		   =					extended_euclid(UpperCAmelCase_    ,			UpperCAmelCase_      )  # Implemented below
 if b < 0:
  UpperCAmelCase			:      Optional[int]		   =					(b % n + n) % n
 return b




def  UpperCamelCase(	UpperCAmelCase_    ,			UpperCAmelCase_    ,			UpperCAmelCase_      ):
 UpperCAmelCase			:      Tuple		   =					invert_modulo(UpperCAmelCase_    ,			UpperCAmelCase_      )
 UpperCAmelCase			:      List[Any]		   =					(b * s) % n
 return x




def  UpperCamelCase(	UpperCAmelCase_    ,			UpperCAmelCase_      ):
 assert a >= 0 and b >= 0

 if b == 0:
  UpperCAmelCase   ,					UpperCAmelCase   ,					UpperCAmelCase			:      Dict		   =					a, 1, 0
 else:
  ((UpperCAmelCase)   ,					(UpperCAmelCase)   ,					(UpperCAmelCase))			:      Optional[int]		   =					extended_gcd(UpperCAmelCase_    ,			a % b      )
  UpperCAmelCase			:      Dict		   =					q
  UpperCAmelCase			:      int		   =					p - q * (a // b)

 assert a % d == 0 and b % d == 0
 assert d == a * x + b * y

 return (d, x, y)




def  UpperCamelCase(	UpperCAmelCase_    ,			UpperCAmelCase_      ):
 if b == 0:
  return (1, 0)
 ((UpperCAmelCase)   ,					(UpperCAmelCase))			:      Tuple		   =					extended_euclid(UpperCAmelCase_    ,			a % b      )
 UpperCAmelCase			:      Tuple		   =					a // b
 return (y, x - k * y)




def  UpperCamelCase(	UpperCAmelCase_    ,			UpperCAmelCase_      ):
 if a < b:
  UpperCAmelCase   ,					UpperCAmelCase			:      Union[str, Any]		   =					b, a

 while a % b != 0:
  UpperCAmelCase   ,					UpperCAmelCase			:      int		   =					b, a % b

 return b


if __name__ == "__main__":
 from doctest import testmod

 testmod(name="modular_division", verbose=True)
 testmod(name="modular_division2", verbose=True)
 testmod(name="invert_modulo", verbose=True)
 testmod(name="extended_gcd", verbose=True)
 testmod(name="extended_euclid", verbose=True)
 testmod(name="greatest_common_divisor", verbose=True)




