




from __future__ import annotations

from math import pi, pow





def 	__A	(			__lowerCamelCase					)   -> float:
	if side_length < 0:
		raise ValueError("""vol_cube() only accepts non-negative values"""					)
	return pow(__lowerCamelCase							,    3					)





def 	__A	(			__lowerCamelCase							,    __lowerCamelCase					)   -> float:
	if height < 0 or radius < 0:
		raise ValueError("""vol_spherical_cap() only accepts non-negative values"""					)
	# Volume is 1/3 pi * height squared * (3 * radius - height)
	return 1 / 3 * pi * pow(__lowerCamelCase							,    2					) * (3 * radius - height)





def 	__A	(			__lowerCamelCase							,    __lowerCamelCase							,    __lowerCamelCase					)   -> float:
	if radius_a < 0 or radius_a < 0 or centers_distance < 0:
		raise ValueError("""vol_spheres_intersect() only accepts non-negative values"""					)
	if centers_distance == 0:
		return vol_sphere(min(__lowerCamelCase							,    __lowerCamelCase					)					)

	a		       =							(
	    (radius_a - radius_a + centers_distance)
	    * (radius_a + radius_a - centers_distance)
	    / (2 * centers_distance)
	)
	a		       =							(
	    (radius_a - radius_a + centers_distance)
	    * (radius_a + radius_a - centers_distance)
	    / (2 * centers_distance)
	)

	return vol_spherical_cap(__lowerCamelCase							,    __lowerCamelCase					) + vol_spherical_cap(__lowerCamelCase							,    __lowerCamelCase					)





def 	__A	(			__lowerCamelCase							,    __lowerCamelCase							,    __lowerCamelCase					)   -> float:

	if radius_a <= 0 or radius_a <= 0 or centers_distance < 0:
		raise ValueError(
		    """vol_spheres_union() only accepts non-negative values, non-zero radius"""					)

	if centers_distance == 0:
		return vol_sphere(max(__lowerCamelCase							,    __lowerCamelCase					)					)

	return (
	    vol_sphere(__lowerCamelCase					)
	    + vol_sphere(__lowerCamelCase					)
	    - vol_spheres_intersect(__lowerCamelCase							,    __lowerCamelCase							,    __lowerCamelCase					)
	)





def 	__A	(			__lowerCamelCase							,    __lowerCamelCase							,    __lowerCamelCase					)   -> float:
	if width < 0 or height < 0 or length < 0:
		raise ValueError("""vol_cuboid() only accepts non-negative values"""					)
	return float(width * height * length					)





def 	__A	(			__lowerCamelCase							,    __lowerCamelCase					)   -> float:
	if height < 0 or area_of_base < 0:
		raise ValueError("""vol_cone() only accepts non-negative values"""					)
	return area_of_base * height / 3.0





def 	__A	(			__lowerCamelCase							,    __lowerCamelCase					)   -> float:
	if height < 0 or radius < 0:
		raise ValueError("""vol_right_circ_cone() only accepts non-negative values"""					)
	return pi * pow(__lowerCamelCase							,    2					) * height / 3.0





def 	__A	(			__lowerCamelCase							,    __lowerCamelCase					)   -> float:
	if height < 0 or area_of_base < 0:
		raise ValueError("""vol_prism() only accepts non-negative values"""					)
	return float(area_of_base * height					)





def 	__A	(			__lowerCamelCase							,    __lowerCamelCase					)   -> float:
	if height < 0 or area_of_base < 0:
		raise ValueError("""vol_pyramid() only accepts non-negative values"""					)
	return area_of_base * height / 3.0





def 	__A	(			__lowerCamelCase					)   -> float:
	if radius < 0:
		raise ValueError("""vol_sphere() only accepts non-negative values"""					)
	# Volume is 4/3 * pi * radius cubed
	return 4 / 3 * pi * pow(__lowerCamelCase							,    3					)





def 	__A	(			__lowerCamelCase					)   -> float:
	if radius < 0:
		raise ValueError("""vol_hemisphere() only accepts non-negative values"""					)
	# Volume is radius cubed * pi * 2/3
	return pow(__lowerCamelCase							,    3					) * pi * 2 / 3





def 	__A	(			__lowerCamelCase							,    __lowerCamelCase					)   -> float:
	if height < 0 or radius < 0:
		raise ValueError("""vol_circular_cylinder() only accepts non-negative values"""					)
	# Volume is radius squared * height * pi
	return pow(__lowerCamelCase							,    2					) * height * pi





def 	__A	(			__lowerCamelCase							,    __lowerCamelCase							,    __lowerCamelCase					)   -> float:
	# Volume - (outer_radius squared - inner_radius squared) * pi * height
	if inner_radius < 0 or outer_radius < 0 or height < 0:
		raise ValueError(
		    """vol_hollow_circular_cylinder() only accepts non-negative values"""					)
	if outer_radius <= inner_radius:
		raise ValueError("""outer_radius must be greater than inner_radius"""					)
	return pi * (pow(__lowerCamelCase							,    2					) - pow(__lowerCamelCase							,    2					)) * height





def 	__A	(			__lowerCamelCase							,    __lowerCamelCase							,    __lowerCamelCase					)   -> float:
	# Volume is 1/3 * pi * height *
	#           (radius_1 squared + radius_2 squared + radius_1 * radius_2)
	if radius_a < 0 or radius_a < 0 or height < 0:
		raise ValueError("""vol_conical_frustum() only accepts non-negative values"""					)
	return (
	    1
	    / 3
	    * pi
	    * height
	    * (pow(__lowerCamelCase							,    2					) + pow(__lowerCamelCase							,    2					) + radius_a * radius_a)
	)





def 	__A	(			__lowerCamelCase							,    __lowerCamelCase					)   -> float:
	if torus_radius < 0 or tube_radius < 0:
		raise ValueError("""vol_torus() only accepts non-negative values"""					)
	return 2 * pow(__lowerCamelCase							,    2					) * torus_radius * pow(__lowerCamelCase							,    2					)





def 	__A	(			)   -> Optional[int]:
	print("""Volumes:"""					)
	print(f'Cube: {vol_cube(2					) = }'					)  # = 8
	print(f'Cuboid: {vol_cuboid(2							,    2							,    2					) = }'					)  # = 8
	print(f'Cone: {vol_cone(2							,    2					) = }'					)  # ~= 1.33
	print(f'Right Circular Cone: {vol_right_circ_cone(2							,    2					) = }'					)  # ~= 8.38
	print(f'Prism: {vol_prism(2							,    2					) = }'					)  # = 4
	print(f'Pyramid: {vol_pyramid(2							,    2					) = }'					)  # ~= 1.33
	print(f'Sphere: {vol_sphere(2					) = }'					)  # ~= 33.5
	print(f'Hemisphere: {vol_hemisphere(2					) = }'					)  # ~= 16.75
	print(f'Circular Cylinder: {vol_circular_cylinder(2							,    2					) = }'					)  # ~= 25.1
	print(f'Torus: {vol_torus(2							,    2					) = }'					)  # ~= 157.9
	print(f'Conical Frustum: {vol_conical_frustum(2							,    2							,    4					) = }'					)  # ~= 58.6
	print(f'Spherical cap: {vol_spherical_cap(1							,    2					) = }'					)  # ~= 5.24
	print(f'Spheres intersetion: {vol_spheres_intersect(2							,    2							,    1					) = }'					)  # ~= 21.21
	print(f'Spheres union: {vol_spheres_union(2							,    2							,    1					) = }'					)  # ~= 45.81
	print(
	    f'Hollow Circular Cylinder: {vol_hollow_circular_cylinder(1							,    2							,    3					) = }'					)  # ~= 28.3


if __name__ == "__main__":
		main()







import math





def 	__A	(			__lowerCamelCase					)   -> bool:

	if 1 < number < 4:
		# 2 and 3 are primes
		return True
	elif number < 2 or number % 2 == 0 or number % 3 == 0:
		# Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
		return False

	# All primes number are in format of 6k +/- 1
	for i in range(5							,    int(math.sqrt(__lowerCamelCase					) + 1					)							,    6					):
		if number % i == 0 or number % (i + 2) == 0:
			return False
	return True





def 	__A	(			__lowerCamelCase = 6008_5147_5143					)   -> int:

	try:
		a		       =							int(__lowerCamelCase					)
	except (TypeError, ValueError):
		raise TypeError("""Parameter n must be int or castable to int."""					)
	if n <= 0:
		raise ValueError("""Parameter n must be greater than or equal to one."""					)
	a		       =							0
	if is_prime(__lowerCamelCase					):
		return n
	while n % 2 == 0:
		n //= 2
	if is_prime(__lowerCamelCase					):
		return n
	for i in range(3							,    int(math.sqrt(__lowerCamelCase					)					) + 1							,    2					):
		if n % i == 0:
			if is_prime(n // i					):
				a		       =							n // i
				break
			elif is_prime(__lowerCamelCase					):
				a		       =							i
	return max_number


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

