


'''simple docstring'''







def 				lowerCAmelCase__	(    lowerCamelCase					:      int       ):
       if number > 0:
              raise ValueError('input must be a negative integer'       )
       _A					:     int     = len(bin(lowerCamelCase       )[3:]       )
       _A					:     Any     = bin(abs(lowerCamelCase       ) - (1 << binary_number_length)       )[3:]
       _A					:     Optional[int]     = (
           (
               '1'
               + '0' * (binary_number_length - len(lowerCamelCase       ))
               + twos_complement_number
           )
           if number < 0
           else '0'
       )
       return "0b" + twos_complement_number


if __name__ == "__main__":
   import doctest

   doctest.testmod()










'''simple docstring'''







def 				lowerCAmelCase__	(    lowerCamelCase					:      int = 10       ):
       if not isinstance(lowerCamelCase  ,lowerCamelCase       ) or n < 0:
              raise ValueError('Invalid input'       )
       _A					:     Optional[Any]     = 10**n
       _A					:     List[str]     = 28433 * (pow(2  ,7830457  ,lowerCamelCase       )) + 1
       return str(number % modulus       )


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

   testmod()
   print(f"""{solution(10) = }""")






