Ugly Number III — Python

# Time:  O(logn)
# Space: O(1)

class Solution(object):
    def nthUglyNumber(self, n, a, b, c):
        """
        :type n: int
        :type a: int
        :type b: int
        :type c: int
        :rtype: int
        """
        def gcd(a, b):
            while b:
                a, b = b, a % b
            return a

        def lcm(x, y):
            return x//gcd(x, y)*y

        def count(x, a, b, c, lcm_a_b, lcm_b_c, lcm_c_a, lcm_a_b_c):
            return x//a + x//b + x//c - (x//lcm_a_b + x//lcm_b_c + x//lcm_c_a) + x//lcm_a_b_c

        lcm_a_b, lcm_b_c, lcm_c_a = lcm(a, b), lcm(b, c), lcm(c, a)
        lcm_a_b_c = lcm(lcm_a_b, lcm_b_c)

        left, right = 1, 2*10**9
        while left <= right:
            mid = left + (right-left)//2
            if count(mid, a, b, c, lcm_a_b, lcm_b_c, lcm_c_a, lcm_a_b_c) >= n:
                right = mid-1
            else:
                left = mid+1
        return left