> For the complete documentation index, see [llms.txt](https://docs.1l.lol/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://docs.1l.lol/number-theory/gcd.md).

# 辗转相除法求 GCD

### 辗转相除法（欧几里得算法）

求最大公约数：`gcd(a, b) = gcd(b, a % b)`，直到 `b = 0`，此时 `a` 即为答案。

```cpp
int gcd(int a, int b) {
    return b == 0 ? a : gcd(b, a % b);   // 递归
}

int gcd2(int a, int b) {                 // 迭代
    while (b) { int t = a % b; a = b; b = t; }
    return a;
}
```

### 最小公倍数

```cpp
int lcm(int a, int b) {
    return a / gcd(a, b) * b;    // 先除后乘，防止溢出
}
```

### 复杂度

每次取模后数字至少减半，复杂度 **O(log min(a,b))**。

### 原理

设 `a = q·b + r`，则 a、b 的公约数集合与 b、r 的公约数集合相同，所以 `gcd(a,b) = gcd(b,r)`。

### STL 用法（C++17）

```cpp
#include <numeric>
int g = std::gcd(a, b);
int l = std::lcm(a, b);
```

### 应用

约分、同余、扩展欧几里得、求解一元线性同余方程等。


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