




'''simple docstring'''


from math import sqrt
def 						lowerCamelCase__   (				a							):
						assert isinstance(a    ,	a							) and (
						    number >= 0
						), "'number' must been an int and positive"

						__snake_case						 =   True

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

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

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

						return status
def 						lowerCamelCase__   (				a							):
						assert isinstance(a    ,	a							) and (n > 2), "'N' must been an int and > 2"

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

						__snake_case						 =   []  # this list will be returns.

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

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

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

						return ans
def 						lowerCamelCase__   (				a							):
						assert isinstance(a    ,	a							) and (n > 2), "'N' must been an int and > 2"

						__snake_case						 =   []

						# 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(a							):
																		ans.append(a							)

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

						return ans
def 						lowerCamelCase__   (				a							):
						assert isinstance(a    ,	a							) and number >= 0, "'number' must been an int and >= 0"

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

						# potential prime number factors.

						__snake_case						 =   2

						__snake_case						 =   number

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

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

						else:
												ans.append(a							)

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

						return ans
def 						lowerCamelCase__   (				a							):
						assert isinstance(a    ,	a							) and (
						    number >= 0
						), "'number' bust been an int and >= 0"

						__snake_case						 =   0

						# prime factorization of 'number'
						__snake_case						 =   prime_factorization(a							)

						__snake_case						 =   max(a							)

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

						return ans
def 						lowerCamelCase__   (				a							):
						assert isinstance(a    ,	a							) and (
						    number >= 0
						), "'number' bust been an int and >= 0"

						__snake_case						 =   0

						# prime factorization of 'number'
						__snake_case						 =   prime_factorization(a							)

						__snake_case						 =   min(a							)

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

						return ans
def 						lowerCamelCase__   (				a							):
						assert isinstance(a    ,	a							), "'number' must been an int"
						assert isinstance(number % 2 == 0    ,	a							), "compare bust been from type bool"

						return number % 2 == 0
def 						lowerCamelCase__   (				a							):
						assert isinstance(a    ,	a							), "'number' must been an int"
						assert isinstance(number % 2 != 0    ,	a							), "compare bust been from type bool"

						return number % 2 != 0
def 						lowerCamelCase__   (				a							):
						assert (
						    isinstance(a    ,	a							) and (number > 2) and is_even(a							)
						), "'number' must been an int, even and > 2"

						__snake_case						 =   []  # this list will returned

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

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

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

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

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

																		j += 1

												i += 1

						# precondition
						assert (
						    isinstance(a    ,	a							)
						    and (len(a							) == 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 						lowerCamelCase__   (				a    ,	a							):
						assert (
						    isinstance(a    ,	a							)
						    and isinstance(a    ,	a							)
						    and (numbera >= 0)
						    and (numbera >= 0)
						), "'number1' and 'number2' must been positive integer."

						__snake_case						 =   0

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

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

						return numbera
def 						lowerCamelCase__   (				a    ,	a							):
						assert (
						    isinstance(a    ,	a							)
						    and isinstance(a    ,	a							)
						    and (numbera >= 1)
						    and (numbera >= 1)
						), "'number1' and 'number2' must been positive integer."

						__snake_case						 =   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'
												__snake_case						 =   prime_factorization(a							)
												__snake_case						 =   prime_factorization(a							)

						elif numbera == 1 or numbera == 1:
												__snake_case						 =   []
												__snake_case						 =   []
												__snake_case						 =   max(a    ,	a							)

						__snake_case						 =   0
						__snake_case						 =   0

						__snake_case						 =   []  # 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:
																								__snake_case						 =   prime_fac_a.count(a							)
																								__snake_case						 =   prime_fac_a.count(a							)

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

																		else:
																								__snake_case						 =   prime_fac_a.count(a							)

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

																		done.append(a							)

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

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

																		done.append(a							)

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

						return ans
def 						lowerCamelCase__   (				a							):
						assert isinstance(a    ,	a							) and (n >= 0), "'number' must been a positive int"

						__snake_case						 =   0
						__snake_case						 =   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(a							):
																		ans += 1

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

						return ans
def 						lowerCamelCase__   (				a    ,	a							):
						assert (
						    is_prime(a							) and is_prime(a							) and (p_number_a < p_number_a)
						), "The arguments must been prime numbers and 'pNumber1' < 'pNumber2'"

						__snake_case						 =   p_number_a + 1  # jump to the next number

						__snake_case						 =   []  # this list will be returns.

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

						while number < p_number_a:
												ans.append(a							)

												number += 1

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

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

						# 'ans' contains not 'pNumber1' and 'pNumber2' !
						return ans
def 						lowerCamelCase__   (				a							):
						assert isinstance(a    ,	a							) and (n >= 1), "'n' must been int and >= 1"

						__snake_case						 =   []  # will be returned.

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

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

						return ans
def 						lowerCamelCase__   (				a							):
						assert isinstance(a    ,	a							) and (
						    number > 1
						), "'number' must been an int and >= 1"

						__snake_case						 =   get_divisors(a							)

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

						# summed all divisors up to 'number' (exclusive), hence [:-1]
						return sum(divisors[:-1]							) == number
def 						lowerCamelCase__   (				a    ,	a							):
						assert (
						    isinstance(a    ,	a							)
						    and isinstance(a    ,	a							)
						    and (denominator != 0)
						), "The arguments must been from type int and 'denominator' != 0"

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

						# precondition
						assert (
						    isinstance(a    ,	a							)
						    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 						lowerCamelCase__   (				a							):
						assert isinstance(a    ,	a							) and (n >= 0), "'n' must been a int and >= 0"

						__snake_case						 =   1  # this will be return.

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

						return ans






def 						lowerCamelCase__   (				a							):
						assert isinstance(a    ,	a							) and (n >= 0), "'n' must been an int and >= 0"

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

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

						return ans







'''simple docstring'''


def 						lowerCamelCase__   (				a							):
						if number < 0:
												raise ValueError('number must not be negative'							)
						return number & (number - 1) == 0


if __name__ == "__main__":
	import doctest

	doctest.testmod()

