AMC 8 2024 Problem 1
AMC8 2024 年第 1 题
表达式与运算★☆☆☆☆
What is the ones digit of 222,222 - 22,222 - 2,222 - 222 - 22 - 2?
请问下方算式的个位数是几？
- A.0
- B.2
- C.4
- D.6
- E.8
正确答案 · Correct Answer
B
解析 · Solution
解法1
我们可以将表达式重写为 222,222-(22,222+2,222+222+22+2) 。我们注意到 22,222+2,222+222+22+2 的个位数字是 0，因为五个数的个位数字都是 2，而 的个位数字是 0。现在，我们有一个个位数字为 0 的数从 222,222 中减去，因此这个表达式的个位数字是 2。
222,222-22,222 = 200,000 200,000 - 2,222 = 197778 197778 = 197556 197556 = 197534 197534 = 197532 所以答案是2。（注意：暴力枚举会花费大量时间，而AMC8是限时考试。）
我们只需要关心个位数。因此，2 以 0 结尾，0 在借位后 (10-2) 以 8 结尾，8 以 6 结尾，6 以 4 结尾，4 以 2 结尾。
我们只需取每个数的个位数然后相减，为了避免得到负数，给第一个数多加10：(12-2)-(2+2+2+2)=10-8=2
我们可以忽略其他数字，只计算 22-2-2-2-2-2 。因为减去五个 2 ，而 ，所以从 22 中减去 10。得到 12，因此末位数字是 2。
Solution 1
We can rewrite the expression as 222,222-(22,222+2,222+222+22+2). We note that the units digit of 22,222+2,222+222+22+2 is 0 because all the units digits of the five numbers are 2 and , which has a units digit of 0. Now, we have something with a units digit of 0 subtracted from 222,222, and so the units digit of this expression is 2.
222,222-22,222 = 200,000 200,000 - 2,222 = 197778 197778 = 197556 197556 = 197534 197534 = 197532 So our answer is 2. (Note that brute forcing takes lots of time and the AMC8 is a timed test)
We only care about the units digits. Thus, 2 ends in 0, 0 after regrouping(10-2) ends in 8, 8 ends in 6, 6 ends in 4, and 4 ends in 2.
We just take the units digit of each and subtract, adding an extra ten to the first number so we don't get a negative number: (12-2)-(2+2+2+2)=10-8=2
We can ignore the other digits and just do 22-2-2-2-2-2. Because you are subtracting five and , you subtract 10 from 22. This gives us 12, so the last digit is 2.
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