奇偶性

考什么 · What it tests

需要先会 · Prerequisites

• 整除与因数（偶数 = 能被 $2$ 整除） / Divisibility (even = divisible by $2$)
• 四则运算 / The four operations
• 分类讨论 / Casework

常见套路 · Common moves

1. 记死运算表：奇 ± 奇 = 偶、奇 ± 偶 = 奇；偶 × 任何 = 偶，只有奇 × 奇才是奇——奇偶题全靠这张表开路。

   Memorize the parity table: odd ± odd = even, odd ± even = odd; even × anything = even, and only odd × odd is odd — every parity problem opens with this table.
2. 用奇偶做「不可能」判定：要证某种凑法办不到，只要算出两边奇偶性不同就直接否决，不必真去凑。

   Use parity to prove impossibility: to show some arrangement can't work, just show the two sides have different parity and reject it — no need to actually construct it.
3. 和的奇偶只数奇数个数：一串数相加，结果的奇偶只看里面有几个奇数——偶数个奇数相加为偶。

   A sum's parity comes only from the odd terms: adding a list, the result's parity depends only on how many odd numbers there are — an even count of odds sums to even.

易错点 · Common pitfalls

• 忘了 0 是偶数：0 能被 2 整除，属于偶数，分类时别漏。

  Forgetting that 0 is even: 0 is divisible by 2 and counts as even — don't drop it in casework.
• 乘法奇偶错判：只要有一个因数是偶数，积就是偶数，别以为「奇数多就是奇数」。

  Misjudging a product's parity: a single even factor makes the product even — "more odds means odd" is false.