

from math import sqrt


def        __lowercase    (  UpperCAmelCase__      ):

					"""simple docstring"""






					assert isinstance(UpperCAmelCase__		,					UpperCAmelCase__      ) and (
					    number >= 0
					), "'number' must been an int and positive"

					__lowerCAmelCase					=							True

					# 0 and 1 are none primes.
					if number <= 1:
										__lowerCAmelCase					=							False

					for divisor in range(2		,					int(round(sqrt(UpperCAmelCase__      )      )      ) + 1      ):
										# if 'number' divisible by 'divisor' then sets 'status'
										# of false and break up the loop.
										if number % divisor == 0:
															__lowerCAmelCase					=							False
															break

    # precondition
					assert isinstance(UpperCAmelCase__		,					UpperCAmelCase__      ), "'status' must been from type bool"

					return status


def        __lowercase    (  UpperCAmelCase__      ):

					"""simple docstring"""






					assert isinstance(UpperCAmelCase__		,					UpperCAmelCase__      ) and (n > 2), "'N' must been an int and > 2"

					# beginList: contains all natural numbers from 2 up to N
					__lowerCAmelCase					=							list(range(2		,					n + 1      )      )

					__lowerCAmelCase					=							[]  # this list will be returns.

					# actual sieve of erathostenes
					for i in range(len(UpperCAmelCase__      )      ):
										for j in range(i + 1		,					len(UpperCAmelCase__      )      ):
															if (begin_list[i] != 0) and (begin_list[j] % begin_list[i] == 0):
																				__lowerCAmelCase					=							0

    # filters actual prime numbers.
					__lowerCAmelCase					=							[x for x in begin_list if x != 0]

					# precondition
					assert isinstance(UpperCAmelCase__		,					UpperCAmelCase__      ), "'ans' must been from type list"

					return ans


def        __lowercase    (  UpperCAmelCase__      ):

					"""simple docstring"""






					assert isinstance(UpperCAmelCase__		,					UpperCAmelCase__      ) and (n > 2), "'N' must been an int and > 2"

					__lowerCAmelCase					=							[]

					# iterates over all numbers between 2 up to N+1
					# if a number is prime then appends to list 'ans'
					for number in range(2		,					n + 1      ):
										if is_prime(UpperCAmelCase__      ):
															ans.append(UpperCAmelCase__      )

    # precondition
					assert isinstance(UpperCAmelCase__		,					UpperCAmelCase__      ), "'ans' must been from type list"

					return ans


def        __lowercase    (  UpperCAmelCase__      ):

					"""simple docstring"""






					assert isinstance(UpperCAmelCase__		,					UpperCAmelCase__      ) and number >= 0, "'number' must been an int and >= 0"

					__lowerCAmelCase					=							[]  # this list will be returns of the function.

					# potential prime number factors.

					__lowerCAmelCase					=							2

					__lowerCAmelCase					=							number

					if number == 0 or number == 1:
										ans.append(UpperCAmelCase__      )

					# if 'number' not prime then builds the prime factorization of 'number'
					elif not is_prime(UpperCAmelCase__      ):
										while quotient != 1:
															if is_prime(UpperCAmelCase__      ) and (quotient % factor == 0):
																				ans.append(UpperCAmelCase__      )
																				quotient /= factor
															else:
																				factor += 1

					else:
										ans.append(UpperCAmelCase__      )

					# precondition
					assert isinstance(UpperCAmelCase__		,					UpperCAmelCase__      ), "'ans' must been from type list"

					return ans


def        __lowercase    (  UpperCAmelCase__      ):

					"""simple docstring"""






					assert isinstance(UpperCAmelCase__		,					UpperCAmelCase__      ) and (
					    number >= 0
					), "'number' bust been an int and >= 0"

					__lowerCAmelCase					=							0

					# prime factorization of 'number'
					__lowerCAmelCase					=							prime_factorization(UpperCAmelCase__      )

					__lowerCAmelCase					=							max(UpperCAmelCase__      )

					# precondition
					assert isinstance(UpperCAmelCase__		,					UpperCAmelCase__      ), "'ans' must been from type int"

					return ans


def        __lowercase    (  UpperCAmelCase__      ):

					"""simple docstring"""






					assert isinstance(UpperCAmelCase__		,					UpperCAmelCase__      ) and (
					    number >= 0
					), "'number' bust been an int and >= 0"

					__lowerCAmelCase					=							0

					# prime factorization of 'number'
					__lowerCAmelCase					=							prime_factorization(UpperCAmelCase__      )

					__lowerCAmelCase					=							min(UpperCAmelCase__      )

					# precondition
					assert isinstance(UpperCAmelCase__		,					UpperCAmelCase__      ), "'ans' must been from type int"

					return ans


def        __lowercase    (  UpperCAmelCase__      ):

					"""simple docstring"""






					assert isinstance(UpperCAmelCase__		,					UpperCAmelCase__      ), "'number' must been an int"
					assert isinstance(number % 2 == 0		,					UpperCAmelCase__      ), "compare bust been from type bool"

					return number % 2 == 0


def        __lowercase    (  UpperCAmelCase__      ):

					"""simple docstring"""






					assert isinstance(UpperCAmelCase__		,					UpperCAmelCase__      ), "'number' must been an int"
					assert isinstance(number % 2 != 0		,					UpperCAmelCase__      ), "compare bust been from type bool"

					return number % 2 != 0


def        __lowercase    (  UpperCAmelCase__      ):

					"""simple docstring"""






					assert (
					    isinstance(UpperCAmelCase__		,					UpperCAmelCase__      ) and (number > 2) and is_even(UpperCAmelCase__      )
					), "'number' must been an int, even and > 2"

					__lowerCAmelCase					=							[]  # this list will returned

					# creates a list of prime numbers between 2 up to 'number'
					__lowerCAmelCase					=							get_prime_numbers(UpperCAmelCase__      )
					__lowerCAmelCase					=							len(UpperCAmelCase__      )

					# run variable for while-loops.
					__lowerCAmelCase					=							0
					__lowerCAmelCase					=							None

					# exit variable. for break up the loops
					__lowerCAmelCase					=							True

					while i < len_pn and loop:
										__lowerCAmelCase					=							i + 1

										while j < len_pn and loop:
															if prime_numbers[i] + prime_numbers[j] == number:
																				__lowerCAmelCase					=							False
																				ans.append(prime_numbers[i]      )
																				ans.append(prime_numbers[j]      )

															j += 1

										i += 1

					# precondition
					assert (
					    isinstance(UpperCAmelCase__		,					UpperCAmelCase__      )
					    and (len(UpperCAmelCase__      ) == 2)
					    and (ans[0] + ans[1] == number)
					    and is_prime(ans[0]      )
					    and is_prime(ans[1]      )
					), "'ans' must contains two primes. And sum of elements must been eq 'number'"

					return ans


def        __lowercase    (  UpperCAmelCase__		,					UpperCAmelCase__      ):

					"""simple docstring"""






					assert (
					    isinstance(UpperCAmelCase__		,					UpperCAmelCase__      )
					    and isinstance(UpperCAmelCase__		,					UpperCAmelCase__      )
					    and (numbera >= 0)
					    and (numbera >= 0)
					), "'number1' and 'number2' must been positive integer."

					__lowerCAmelCase					=							0

					while numbera != 0:
										__lowerCAmelCase					=							numbera % numbera
										__lowerCAmelCase					=							numbera
										__lowerCAmelCase					=							rest

					# precondition
					assert isinstance(UpperCAmelCase__		,					UpperCAmelCase__      ) and (
					    numbera >= 0
					), "'number' must been from type int and positive"

					return numbera


def        __lowercase    (  UpperCAmelCase__		,					UpperCAmelCase__      ):

					"""simple docstring"""






					assert (
					    isinstance(UpperCAmelCase__		,					UpperCAmelCase__      )
					    and isinstance(UpperCAmelCase__		,					UpperCAmelCase__      )
					    and (numbera >= 1)
					    and (numbera >= 1)
					), "'number1' and 'number2' must been positive integer."

					__lowerCAmelCase					=							1  # actual answer that will be return.

					# for kgV (x,1)
					if numbera > 1 and numbera > 1:
										# builds the prime factorization of 'number1' and 'number2'
										__lowerCAmelCase					=							prime_factorization(UpperCAmelCase__      )
										__lowerCAmelCase					=							prime_factorization(UpperCAmelCase__      )

					elif numbera == 1 or numbera == 1:
										__lowerCAmelCase					=							[]
										__lowerCAmelCase					=							[]
										__lowerCAmelCase					=							max(UpperCAmelCase__		,					UpperCAmelCase__      )

					__lowerCAmelCase					=							0
					__lowerCAmelCase					=							0

					__lowerCAmelCase					=							[]  # captured numbers int both 'primeFac1' and 'primeFac2'

					# iterates through primeFac1
					for n in prime_fac_a:
										if n not in done:
															if n in prime_fac_a:
																				__lowerCAmelCase					=							prime_fac_a.count(UpperCAmelCase__      )
																				__lowerCAmelCase					=							prime_fac_a.count(UpperCAmelCase__      )

																				for _ in range(max(UpperCAmelCase__		,					UpperCAmelCase__      )      ):
																									ans *= n

															else:
																				__lowerCAmelCase					=							prime_fac_a.count(UpperCAmelCase__      )

																				for _ in range(UpperCAmelCase__      ):
																									ans *= n

															done.append(UpperCAmelCase__      )

    # iterates through primeFac2
					for n in prime_fac_a:
										if n not in done:
															__lowerCAmelCase					=							prime_fac_a.count(UpperCAmelCase__      )

															for _ in range(UpperCAmelCase__      ):
																				ans *= n

															done.append(UpperCAmelCase__      )

    # precondition
					assert isinstance(UpperCAmelCase__		,					UpperCAmelCase__      ) and (
					    ans >= 0
					), "'ans' must been from type int and positive"

					return ans


def        __lowercase    (  UpperCAmelCase__      ):

					"""simple docstring"""






					assert isinstance(UpperCAmelCase__		,					UpperCAmelCase__      ) and (n >= 0), "'number' must been a positive int"

					__lowerCAmelCase					=							0
					__lowerCAmelCase					=							2  # this variable holds the answer

					while index < n:
										index += 1

										ans += 1  # counts to the next number

										# if ans not prime then
										# runs to the next prime number.
										while not is_prime(UpperCAmelCase__      ):
															ans += 1

    # precondition
					assert isinstance(UpperCAmelCase__		,					UpperCAmelCase__      ) and is_prime(
					    UpperCAmelCase__      ), "'ans' must been a prime number and from type int"

					return ans


def        __lowercase    (  UpperCAmelCase__		,					UpperCAmelCase__      ):

					"""simple docstring"""






					assert (
					    is_prime(UpperCAmelCase__      ) and is_prime(UpperCAmelCase__      ) and (p_number_a < p_number_a)
					), "The arguments must been prime numbers and 'pNumber1' < 'pNumber2'"

					__lowerCAmelCase					=							p_number_a + 1  # jump to the next number

					__lowerCAmelCase					=							[]  # this list will be returns.

					# if number is not prime then
					# fetch the next prime number.
					while not is_prime(UpperCAmelCase__      ):
										number += 1

					while number < p_number_a:
										ans.append(UpperCAmelCase__      )

										number += 1

										# fetch the next prime number.
										while not is_prime(UpperCAmelCase__      ):
															number += 1

    # precondition
					assert (
					    isinstance(UpperCAmelCase__		,					UpperCAmelCase__      )
					    and ans[0] != p_number_a
					    and ans[len(UpperCAmelCase__      ) - 1] != p_number_a
					), "'ans' must been a list without the arguments"

					# 'ans' contains not 'pNumber1' and 'pNumber2' !
					return ans


def        __lowercase    (  UpperCAmelCase__      ):

					"""simple docstring"""






					assert isinstance(UpperCAmelCase__		,					UpperCAmelCase__      ) and (n >= 1), "'n' must been int and >= 1"

					__lowerCAmelCase					=							[]  # will be returned.

					for divisor in range(1		,					n + 1      ):
										if n % divisor == 0:
															ans.append(UpperCAmelCase__      )

    # precondition
					assert ans[0] == 1 and ans[len(UpperCAmelCase__      ) - 1] == n, "Error in function getDivisiors(...)"

					return ans


def        __lowercase    (  UpperCAmelCase__      ):

					"""simple docstring"""






					assert isinstance(UpperCAmelCase__		,					UpperCAmelCase__      ) and (
					    number > 1
					), "'number' must been an int and >= 1"

					__lowerCAmelCase					=							get_divisors(UpperCAmelCase__      )

					# precondition
					assert (
					    isinstance(UpperCAmelCase__		,					UpperCAmelCase__      )
					    and (divisors[0] == 1)
					    and (divisors[len(UpperCAmelCase__      ) - 1] == number)
					), "Error in help-function getDivisiors(...)"

					# summed all divisors up to 'number' (exclusive), hence [:-1]
					return sum(divisors[:-1]      ) == number


def        __lowercase    (  UpperCAmelCase__		,					UpperCAmelCase__      ):

					"""simple docstring"""






					assert (
					    isinstance(UpperCAmelCase__		,					UpperCAmelCase__      )
					    and isinstance(UpperCAmelCase__		,					UpperCAmelCase__      )
					    and (denominator != 0)
					), "The arguments must been from type int and 'denominator' != 0"

					# build the greatest common divisor of numerator and denominator.
					__lowerCAmelCase					=							gcd(abs(UpperCAmelCase__      )		,					abs(UpperCAmelCase__      )      )

					# precondition
					assert (
					    isinstance(UpperCAmelCase__		,					UpperCAmelCase__      )
					    and (numerator % gcd_of_fraction == 0)
					    and (denominator % gcd_of_fraction == 0)
					), "Error in function gcd(...,...)"

					return (numerator // gcd_of_fraction, denominator // gcd_of_fraction)


def        __lowercase    (  UpperCAmelCase__      ):

					"""simple docstring"""






					assert isinstance(UpperCAmelCase__		,					UpperCAmelCase__      ) and (n >= 0), "'n' must been a int and >= 0"

					__lowerCAmelCase					=							1  # this will be return.

					for factor in range(1		,					n + 1      ):
										ans *= factor

					return ans


def        __lowercase    (  UpperCAmelCase__      ):

					"""simple docstring"""






					assert isinstance(UpperCAmelCase__		,					UpperCAmelCase__      ) and (n >= 0), "'n' must been an int and >= 0"

					__lowerCAmelCase					=							0
					__lowerCAmelCase					=							1
					__lowerCAmelCase					=							1  # this will be return

					for _ in range(n - 1      ):
										__lowerCAmelCase					=							ans
										ans += fiba
										__lowerCAmelCase					=							tmp

					return ans









from graphs.minimum_spanning_tree_kruskal import kruskal


def        __lowercase    (  ):

					"""simple docstring"""






					__lowerCAmelCase					=							9
					__lowerCAmelCase					=							[
					    [0, 1, 4],
					    [0, 7, 8],
					    [1, 2, 8],
					    [7, 8, 7],
					    [7, 6, 1],
					    [2, 8, 2],
					    [8, 6, 6],
					    [2, 3, 7],
					    [2, 5, 4],
					    [6, 5, 2],
					    [3, 5, 14],
					    [3, 4, 9],
					    [5, 4, 10],
					    [1, 7, 11],
					]

					__lowerCAmelCase					=							kruskal(UpperCAmelCase__		,					UpperCAmelCase__      )

					__lowerCAmelCase					=							[
					    [7, 6, 1],
					    [2, 8, 2],
					    [6, 5, 2],
					    [0, 1, 4],
					    [2, 5, 4],
					    [2, 3, 7],
					    [0, 7, 8],
					    [3, 4, 9],
					]

					assert sorted(UpperCAmelCase__      ) == sorted(UpperCAmelCase__      )






