AMC8 2022

AMC8 2022 · Q17

AMC8 2022 · Q17. It mainly tests Digit properties (sum of digits, divisibility tests), Powers & residues.

If $n$ is an even positive integer, the $\text{double factorial}$ notation $n!!$ represents the product of all the even integers from $2$ to $n$. For example, $8!! = 2 \cdot 4 \cdot 6 \cdot 8$. What is the units digit of the following sum? \[2!! + 4!! + 6!! + \cdots + 2018!! + 2020!! + 2022!!\]

如果$n$是偶正整数，双阶乘记号$n!!$表示从$2$到$n$的所有偶整数的乘积。例如，$8!! = 2 \cdot 4 \cdot 6 \cdot 8$。以下和的单位数是多少？ \[2!! + 4!! + 6!! + \cdots + 2018!! + 2020!! + 2022!!\]

(A) 0 0

(B) 2 2

(D) 6 6

(E) 8 8

Answer

Correct choice: (B)

正确答案：（B）

Solution

Notice that once $n>8,$ the units digit of $n!!$ will be $0$ because there will be a factor of $10.$ Thus, we only need to calculate the units digit of \[2!!+4!!+6!!+8!! = 2+8+48+48\cdot8.\] We only care about units digits, so we have $2+8+8+8\cdot8,$ which has the same units digit as $2+8+8+4.$ The answer is $\boxed{\textbf{(B) } 2}.$

注意到一旦$n>8$，$n!!$的单位数将是$0$，因为会有$10$的因子。因此，我们只需计算 \[2!!+4!!+6!!+8!! = 2+8+48+48\cdot8.\] 的单位数。我们只关心单位数，所以有$2+8+8+8\cdot8$，其单位数与$2+8+8+4$相同。答案是$\boxed{\textbf{(B) } 2}$。

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