> 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/elementary-number-theory.md).

# 初等数论

### 素数与合数

* **素数**：大于 1 且只有 1 和它本身两个约数的数，如 2、3、5、7。
* **合数**：大于 1 且不是素数的数。**1 既不是素数也不是合数**。
* 2 是唯一偶素数。

### 最大公约数与最小公倍数

* **gcd(a, b)**：最大公约数
* **lcm(a, b)**：最小公倍数，`lcm(a,b) = a / gcd(a,b) * b`（先除后乘防溢出）

### 约数与倍数

* 若 `a % b == 0`，则 b 是 a 的约数（因数），a 是 b 的倍数。
* 一个数约数成对出现，枚举约数只需到 `√n`。

```cpp
// 求 n 的所有约数
for (int i = 1; i * i <= n; i++) {
    if (n % i == 0) {
        cout << i << " ";
        if (i != n / i) cout << n / i << " ";
    }
}
```

### 同余与模运算

* `a ≡ b (mod m)` 表示 a、b 除以 m 余数相同。
* 运算规则：`(a + b) % m = (a%m + b%m) % m`，减法、乘法同理。
* **除法不满足**，需要求逆元。
* 取模常配合防溢出：`(a * b) % m`。

### 奇偶性

* 奇 ± 奇 = 偶，偶 ± 偶 = 偶，奇 ± 偶 = 奇
* 奇 × 奇 = 奇，其余乘法结果为偶


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