p033.py

import fractions


def compute():
	# Consider an arbitrary fraction n/d:
	#   Let n = 10 * n1 + n0 be the numerator.
	#   Let d = 10 * d1 + d0 be the denominator.
	# As stated in the problem, we need 10 <= n < d < 100.
	# We must disregard trivial simplifications where n0 = d0 = 0.
	# 
	# Now, a simplification with n0 = d0 is impossible because:
	#   n1 / d1 = n / d = (10*n1 + n0) / (10*d1 + n0).
	#   n1 * (10*d1 + n0) = d1 * (10*n1 + n0).
	#   10*n1*d1 + n1*n0 = 10*d1*n1 + d1*n0.
	#   n1*n0 = d1*n0.
	#   n1 = d1.
	#   This implies n = d, which contradicts the fact that n < d.
	# Similarly, we cannot have a simplification with n1 = d1 for the same reason.
	# 
	# Therefore we only need to consider the cases where n0 = d1 or n1 = d0.
	# In the first case, check that n1/d0 = n/d;
	# in the second case, check that n0/d1 = n/d.
	numer = 1
	denom = 1
	for d in range(10, 100):
		for n in range(10, d):
			n0 = n % 10
			n1 = n // 10
			d0 = d % 10
			d1 = d // 10
			if (n1 == d0 and n0 * d == n * d1) or (n0 == d1 and n1 * d == n * d0):
				numer *= n
				denom *= d
	return str(denom // fractions.gcd(numer, denom))


if __name__ == "__main__":
	print(compute())