
'''simple docstring'''
def    _UpperCamelCase						(       SCREAMING_SNAKE_CASE__ ,       SCREAMING_SNAKE_CASE__ ,       SCREAMING_SNAKE_CASE__			)  ->							Optional[int]:


		'''simple docstring'''


		if n == 0:
				return 1

		elif n % 2 == 1:
				return (binary_exponentiation(SCREAMING_SNAKE_CASE__ ,       n - 1 ,       SCREAMING_SNAKE_CASE__			) * a) % mod

		else:
				snake_case  :       Dict     			=						binary_exponentiation(SCREAMING_SNAKE_CASE__ ,       n / 2 ,       SCREAMING_SNAKE_CASE__			)
				return (b * b) % mod


# a prime number
lowercase__						       =      7_0_1

lowercase__						       =      1_0_0_0_0_0_0_0_0_0
lowercase__						       =      1_0

# using binary exponentiation function, O(log(p)):
print((a / b) % p == (a * binary_exponentiation(b, p - 2, p)) % p)




print((a / b) % p == (a * b ** (p - 2)) % p)




'''simple docstring'''
from __future__ import annotations

from math import ceil, floor, sqrt




def    _UpperCamelCase						(       SCREAMING_SNAKE_CASE__ = 200_0000			)  ->							int:


		'''simple docstring'''


		snake_case  :       list[int]     			=						[0]
		snake_case  :       int

		for idx in range(1 ,       ceil(sqrt(target * 2			) * 1.1			)			):
				triangle_numbers.append(triangle_numbers[-1] + idx			)

		# we want this to be as close as possible to target
		snake_case  :       int     			=						0
		# the area corresponding to the grid that gives the product closest to target
		snake_case  :       int     			=						0
		# an estimate of b, using the quadratic formula
		snake_case  :       float
		# the largest integer less than b_estimate
		snake_case  :       int
		# the largest integer less than b_estimate
		snake_case  :       int
		# the triangle number corresponding to b_floor
		snake_case  :       int
		# the triangle number corresponding to b_ceil
		snake_case  :       int

		for idx_a, triangle_a in enumerate(triangle_numbers[1:] ,       1			):
				snake_case  :       int     			=						(-1 + sqrt(1 + 8 * target / triangle_a			)) / 2
				snake_case  :       Optional[int]     			=						floor(SCREAMING_SNAKE_CASE__			)
				snake_case  :       int     			=						ceil(SCREAMING_SNAKE_CASE__			)
				snake_case  :       Optional[Any]     			=						triangle_numbers[b_floor]
				snake_case  :       Union[str, Any]     			=						triangle_numbers[b_ceil]

				if abs(target - triangle_b_first_guess * triangle_a			) < abs(
				    target - best_product			):
						snake_case  :       str     			=						triangle_b_first_guess * triangle_a
						snake_case  :       int     			=						idx_a * b_floor

				if abs(target - triangle_b_second_guess * triangle_a			) < abs(
				    target - best_product			):
						snake_case  :       List[Any]     			=						triangle_b_second_guess * triangle_a
						snake_case  :       Any     			=						idx_a * b_ceil

		return area


if __name__ == "__main__":
						print(f"{solution() = }")


