AMC 8 2020 Problem 12

AMC8 2020 年第 12 题

整除与因数★★★☆☆

For a positive integer

n

, the factorial notation n! represents the product of the integers from

n

to 1 . What value of

N

satisfies the following equation?

5!\cdot 9!=12\cdot N!

对一个正整数 n，符号 n！表示从 n 到 1 的所有整数的乘积。例如， 6！=6·5·4·3·2·3.2·1 则 N 为何值时，满足下面的方程？

正确答案 · Correct Answer

解析 · Solution

Solution 1

We have

5! = 2 \cdot 3 \cdot 4 \cdot 5

, and

2 \cdot 5 \cdot 9! = 10 \cdot 9! = 10!

. Therefore, the equation becomes

3 \cdot 4 \cdot 10! = 12 \cdot N!

, and so

12 \cdot 10! = 12 \cdot N!

. Cancelling the 12 s, it is clear that N=10. Since 5! = 120, we obtain

120\cdot 9!=12\cdot N!

, which becomes

12\cdot 10\cdot 9!=12\cdot N!

and thus

12 \cdot 10!=12\cdot N!

. We therefore deduce N=10. We can see that the answers

\textbf{(B)}

\textbf{(E)}

contain a factor of 11, but there is no such factor of 11 in

5! \cdot 9!

. The factor 11 is in every answer choice except 10, so four of the possible answers are eliminated. Therefore, the answer must be 10. by HW73

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