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# 1201. Ugly Number III

Write a program to find the `n`-th ugly number.

Ugly numbers are **positive integers** which are divisible by `a` **or** `b` **or** `c`.

**Example 1:**

```
Input: n = 3, a = 2, b = 3, c = 5
Output: 4
Explanation: The ugly numbers are 2, 3, 4, 5, 6, 8, 9, 10... The 3rd is 4.
```

**Example 2:**

```
Input: n = 4, a = 2, b = 3, c = 4
Output: 6
Explanation: The ugly numbers are 2, 3, 4, 6, 8, 9, 10, 12... The 4th is 6.
```

**Example 3:**

```
Input: n = 5, a = 2, b = 11, c = 13
Output: 10
Explanation: The ugly numbers are 2, 4, 6, 8, 10, 11, 12, 13... The 5th is 10.
```

**Example 4:**

```
Input: n = 1000000000, a = 2, b = 217983653, c = 336916467
Output: 1999999984
```

**Constraints:**

* `1 <= n, a, b, c <= 10^9`
* `1 <= a * b * c <= 10^18`
* It's guaranteed that the result will be in range `[1, 2 * 10^9]`

找第N个只能被a或b或c整除的独立的数。暴力法是不断遍历和比较a *a* a ..., b *b ... 和c* c ...进一步优化O(N)=>二分所有整数范围，对于mid，小于a，b和c个数分别为mid / a, mid/b和mid/c。但它们之间有重复，根据简单的集合论，要去除mid/ab，mid/ac和mid/bc再加上mid/abc。当abc并非互质时，去除mid/ab等还不够，应当以ab（其它乘数同理）的最小公倍数为基础去除。

```cpp
/*
 * @lc app=leetcode id=1201 lang=cpp
 *
 * [1201] Ugly Number III
 */

// @lc code=start
class Solution {
public:
    int nthUglyNumber(int n, long a, long b, long c) {
        long start = 1, end = INT_MAX;
        long ab = lcm(a, b), ac = lcm(a, c), bc = lcm(b, c), abc = lcm(a, bc);
        while (start < end) {
            const long mid = start + (end - start) / 2;
            const long cnt = mid / a + mid / b + mid / c - mid / ab - mid / ac - mid / bc + mid / abc;
            if (cnt < n) start = mid + 1;
            else end = mid;
        } 
        return start;
    }
};
// @lc code=end


```