> 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/unique-factorization.md).

# 唯一分解定理

### 定理内容

任何一个大于 1 的整数都可以**唯一**地分解为若干素数之积：

`n = p1^a1 × p2^a2 × ... × pk^ak`

其中 pi 为素数，指数 ai 为正整数。这个分解称为**质因数分解**。

### 质因数分解算法

从小到大试除，最多试到 `√n`：

```cpp
void factorize(int n) {
    for (int i = 2; i * i <= n; i++) {
        while (n % i == 0) {
            cout << i << " ";  // i 是 n 的一个质因子
            n /= i;
        }
    }
    if (n > 1) cout << n << " ";  // 剩下的大素数
}
```

### 用途

* 求约数个数：若 `n = p1^a1 × ... × pk^ak`，则约数个数为 `(a1+1)(a2+1)...(ak+1)`
* 求约数之和：`(1+p1+...+p1^a1) × ... × (1+pk+...+pk^ak)`
* 判断完全平方数、求最大公约数等。

### 示例

`60 = 2² × 3 × 5`，约数个数 = (2+1)×(1+1)×(1+1) = 12。


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