AMC 8 2011 Problem 24
AMC8 2011 年第 24 题
奇偶性★★★★☆
In how many ways can 10001 be written as the sum of two primes?
有多少种方法可以将 10001 写成两个质数之和?
- A.0
- B.1
- C.2
- D.3
- E.4
正确答案 · Correct Answer
A
解析 · Solution
解法1
常规解法 · 奇偶
两质数之和为奇数 10001，必是一奇一偶。唯一的偶质数是 2，故只能写成 2+9999。但 9999=3×3333 能被 3 整除，不是质数。所以一种也没有，答案 0。
巧解 · 哥德巴赫式判据
一个奇数能写成两质数之和，当且仅当它比某个质数大 2（因其中必含偶质数 2）。这里 10001-2=9999，各位数字和为 36 能被 3 整除，9999 是合数。故 10001 无法写成两质数之和，方法数为 0。
Solution 1
Elementary · parity
An odd sum of 10001 needs one even prime and one odd prime; the only even prime is 2, so the only candidate is 2+9999. But is divisible by 3, hence not prime. So there are 0 ways.
Slick · Goldbach-style test
An odd number is a sum of two primes only if it is 2 more than a prime (the even prime 2 must be one of them). Here 10001-2=9999 has digit sum 36, divisible by 3, so 9999 is composite. Thus there are 0 ways.
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