Solution count for equation 1/x + 1/y = 1/n

from time import time
st = time()

goal = 1000
n = goal
solutions = 0

while solutions <= goal:
    solutions = 0
    for x in range(n + 1, 2*n + 1):
        y, rem = divmod(x * n, x - n)
        if rem == 0:
        if (x * n) % (x - n) == 0:
            solutions += 1

    if n % 1000 == 0:
        print('n = {:5}, solutions: {:4}, time is {:6.3f}'.format(n, solutions, time() - st))
    n += 1