AMC8 2020

AMC8 2020 · Q12

AMC8 2020 · Q12. It mainly tests Manipulating equations, Fractions.

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? \[5!\cdot 9!=12\cdot N!\]

对于正整数 $n$，阶乘记号 $n!$ 表示从 $n$ 到 $1$ 的整数乘积。哪一个 $N$ 满足以下方程？\[5!\cdot 9!=12\cdot N!\]

(A) 10 10

(B) 11 11

(D) 13 13

(E) 14\qquad 14\qquad

Answer

Correct choice: (A)

正确答案：（A）

Solution

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 $N=\boxed{\textbf{(A) }10}$.

我们有 $5! = 2 \cdot 3 \cdot 4 \cdot 5$，并且 $2 \cdot 5 \cdot 9! = 10 \cdot 9! = 10!$。因此，方程变为 $3 \cdot 4 \cdot 10! = 12 \cdot N!$，于是 $12 \cdot 10! = 12 \cdot N!$。消去 $12$，显然 $N=\boxed{\textbf{(A) }10}$。

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