AMC 8 2022 Problem 17
AMC8 2022 年第 17 题
余数与同余★★★☆☆
If n is an even positive integer, the double factorial notation n!! represents the product of all the even integers from 2 to n . For example, . What is the units digit of the following sum?
如果 n 是一个正整数，那么双阶乘记号 n！！代表从 2 到 n 的所有偶整数的乘积。例如， 8!!=2·4·6·8。问下面和式的个位数字是几？
2！！+4！！+6！！+ … +2018！！+2020！！+2022！！
- A.0
- B.2
- C.4
- D.6
- E.8
正确答案 · Correct Answer
B
解析 · Solution
Solution 1
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 We only care about units digits, so we have which has the same units digit as 2+8+8+4. The answer is 2. We can see that after 8!! in the sequence, the units digit is always 0 (every value after 8!! is a multiple of 10 ). Therefore, our answer is the sum of the units digits of 2!!, 4!!, 6!!, and 8!! respectively. This sum is equal to 2 + 8 + 8 + 4, or 2.
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