from math import sqrt
    from random import randint
     
    # returns the greatest divisor and
    # how many times it goes into num as a list
    def gd(num):
            if is_prime(num):
                    return [1, num]
            else:
                    i = round(num / 2) + 1
     
                    while i >= 2:
                            for n in range(2, i + 1):
                                    if i * n == num:
                                            return [i, n]
                            i -= 1
     
    def list_check(f, nums):
            for i in nums:
                    if not f(i):
                            return False
     
            return True
     
    # fermat's primality test, runs ten times
    def is_prime(x):
            test_num = 0
     
            for i in range(0, 11):
                    test_num = randint(1, x)
                    if (test_num ** x % x) == (test_num % x):
                            continue
                    else:
                            return False
     
            return True
     
    def simplify(problem):
            nums = {}
     
            for i in problem:
                    nums[i] = 0
     
            for i in problem:
                    if i != 1:
                            nums[i] += 1
     
            return nums
     
    def divisor_count(num):
            nums = {}
            problem = []
            current_divisors = []
     
            current_divisors = gd(num)
            for i in current_divisors:
                    problem.append(i)
     
            while(list_check(is_prime, problem) == False):
                    for i in range(0, len(problem)):
                            if not is_prime(problem[i]):
                                    current_divisors = gd(problem[i])
                                    problem[i] = current_divisors[0]
                                    problem.append(current_divisors[1])
     
            nums = simplify(problem)
     
            divisors = 1
            for k, v in nums.items():
                    divisors *= v + 1
     
            return divisors
     
    def sum(lower, next, f, upper, total):
            if lower > upper:
                    return total
            else:
                    sum(next(lower), next, f, upper, total + f(lower))
     
    def pe_12(num, divisors):
            i = 0
            while(divisors < 500):
                    if i % 10 == 0:
                            print(divisors)
                    i += 1
                    num += 1
                    divisors = divisor_count(triangle_num(num))
            print(num)
     
    def triangle_num(x):
            sum = 0
     
            while x > 0:
                    sum += x
                    x -= 1
     
            return sum