Factor Reactor!

Python 2.7:

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from math import sqrt, factorial
def factors(n):
    if n == 1:
        return 1
    count = 2
    for test in xrange(2, n/2+1):
        if n % test == 0:
            count += 1
    return count
def product(numbers):
    prod = 1
    for number in numbers: prod *= number
    return prod
answer = sqrt(product([n for n in xrange(3, 219) if factors(n) % 2 != 0]))
n = 0
while factorial(n) != answer:
    n += 1
print "Answer: {0}!".format(n)

Numbers with odd factors are squares. $\sqrt{4 * 9 * 16 * ... * 196}$ = 2 * 3 * 4 * ... * 14 = 14! = 87178291200