AMC8 2011

AMC8 2011 · Q24

AMC8 2011 · Q24. It mainly tests Primes & prime factorization, Parity (odd/even).

In how many ways can 10,001 be written as the sum of two primes?

10001可以用两个质数之和表示多少种方式？

(A) 0 0

(B) 1 1

(D) 3 3

(E) 4 4

Answer

Correct choice: (A)

正确答案：（A）

Solution

For the sum of two numbers to be odd, one must be odd and the other must be even, because all odd numbers are of the form $2n+1$ where n is an integer, and all even numbers are of the form $2m$ where m is an integer. \[2n + 1 + 2m = 2m + 2n + 1 = 2(m+n) + 1\] and $m+n$ is an integer because $m$ and $n$ are both integers. The only even prime number is $2,$ so our only combination could be $2$ and $9999.$ But, $9999$ is clearly divisible by $3$, so the number of ways $10001$ can be written as the sum of two primes is $\boxed{\textbf{(A)}\ 0}$.

两个数之和为奇数，一个必须奇一个偶，因为奇数形式$2n+1$，偶数$2m$。 \[2n + 1 + 2m = 2m + 2n + 1 = 2(m+n) + 1\] 唯一偶质数是$2$，所以只有$2+9999$。但$9999$能被$3$整除，所以方式数是$\boxed{\textbf{(A)}\ 0}$。

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