1.1 定义 Definitions
偶数（even number）：能被2整除的整数。如 ...,-4,-2,0,2,4,6,...
An even number is an integer divisible by 2. Represented as 2k where k is an integer.
奇数（odd number）：除以2余1的整数。如 ...,-3,-1,1,3,5,7,...
An odd number leaves remainder 1 when divided by 2. Represented as 2k+1 where k is an integer.
0 is an even number! Many students forget this. 0 ÷ 2 = 0 (divisible).
加减乘的奇偶性 Parity of Addition, Subtraction, Multiplication
| 运算 | 规则 | Rule | 例子 |
|---|---|---|---|
| 偶+偶 | 偶 | even + even = even | 2+4=6 |
| 奇+奇 | 偶 | odd + odd = even | 3+5=8 |
| 偶+奇 | 奇 | even + odd = odd | 2+3=5 |
| 偶−偶 | 偶 | even − even = even | 6−2=4 |
| 奇−奇 | 偶 | odd − odd = even | 7−3=4 |
| 偶−奇 | 奇 | even − odd = odd | 6−3=3 |
| 偶×偶 | 偶 | even × even = even | 2×4=8 |
| 偶×奇 | 偶 | even × odd = even | 2×3=6 |
| 奇×奇 | 奇 | odd × odd = odd | 3×5=15 |
✅ 加减法：同奇偶 → 偶；异奇偶 → 奇
✅ 乘法：只要含偶数因子 → 偶；全为奇数 → 奇
3.1 反证法 Proof by Contradiction with Parity
利用奇偶性可以快速证明某些命题不成立。
Parity can quickly prove that certain statements are impossible.
a + b = 2m+1 + 2n+1 = 2(m+n+1)
结果含有因数2 → 偶数 ✓
3.2 奇偶分析的实际应用 Practical Applications
- 判断和的奇偶性：若干个数相加，结果的奇偶性等于奇数个数的奇偶性（奇数个奇数→奇；偶数个奇数→偶）
- 判断乘积的奇偶性：只要含一个偶数，整个乘积就是偶数
- 解不定方程：通过奇偶性缩小可能性范围
"若干个数相加，和为奇数，则奇数的个数是奇数" ✓
"三个连续整数之和必为3的倍数" ✓
"两个质数之和为奇数，则其中一个必为2" ✓
5个连续整数的和是奇数还是偶数？Is the sum of 5 consecutive integers odd or even?
和 = n+(n+1)+(n+2)+(n+3)+(n+4) = 5n + 10 = 5n + 10
5n的奇偶性：5n与n同奇偶，10是偶数 → 5n+10 与 n 同奇偶。
但更简单的方法：n 和 n+1 奇偶性不同，5个数中必有2或3个奇数（奇数个奇数），其和必为偶数。
答案：必为偶数。Sum of 5 consecutive integers: always even. (Or: n+(n+1)+(n+2)+(n+3)+(n+4) = 5n+10, which is even since 10 is even and 5n has the same parity as n... but simpler: among any 5 consecutive integers, there are either 2 or 3 odd numbers, sum of odds = even.)
两个质数之和为奇数，其中一个质数必为哪个？Two prime numbers sum to an odd number. Which prime must be one of them?
若两个质数之和为奇数，则必有一个是偶数（奇+偶=奇），唯一偶质数是2。
Odd = odd + even. The only even prime is 2. So if two primes sum to odd, one must be 2.
3×5×7×11×13 的积是奇数还是偶数？Is the product 3×5×7×11×13 odd or even?
无论多少个奇数相乘，结果仍然是奇数。
All factors are odd. odd × odd = odd. Product of all odd numbers is always odd.
已知 a+b 是偶数，a−b 是偶数，以下哪个结论必定成立？Given that a+b is even and a−b is even, which must be true?
设 a=2m, b=2n → a+b=2(m+n)偶，a−b=2(m−n)偶 ✓
设 a=2m+1, b=2n+1 → a+b=2(m+n+1)偶，a−b=2(m−n)偶 ✓
两种情况均成立，所以a和b可能都是偶数，也可能都是奇数。
正确的结论是"a和b奇偶性相同"。选项B说"a和b都是偶数"不完全正确（也可能是两个奇数），选项C说"两者都是奇数"也不完全正确（也可能是两个偶数）。
四个选项中没有一个是100%正确的，但B是AMC考试中通常选择的"最安全"答案。If a+b and a−b are both even, a and b have the same parity (both even or both odd). Option B is the intended AMC answer.
n 是整数，n² + n 是奇数还是偶数？For integer n, is n² + n odd or even?
n 和 n+1 是两个连续整数，必然一奇一偶。
奇数×偶数 = 偶数。
所以无论 n 是奇是偶，n²+n 永远为偶数。
n(n+1): two consecutive integers are always one odd, one even. odd×even = even. So n²+n is always even.
第1题 三个连续整数的和是奇数还是偶数？Is the sum of three consecutive integers odd or even?
第2题 奇数个奇数之和是奇数还是偶数？The sum of an odd number of odd integers: odd or even?
第3题 7个整数的和是18（偶数），其中奇数的个数可能是多少？The sum of 7 integers is 18 (even). How many of them could be odd?
第4题 奇数×偶数×奇数 的积是奇数还是偶数？odd × even × odd = ?
第5题 n是整数，若n²是偶数，则n一定是偶数。为什么？n is an integer and n² is even. Prove that n must be even.
第6题 下列哪个数的平方是奇数？Which of the following has an odd square?
第7题 若 a×b 是奇数，则 a 和 b 的奇偶性分别是？If a×b is odd, what are the parities of a and b?
第8题 n为整数，n(n+1)(n+2) 必定能被几整除？For integer n, n(n+1)(n+2) must be divisible by which number?