2014 AMC 8 第 8 题
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所有题目均经美国数学协会（MAA）官方合法授权使用。
8.
初中数学俱乐部的十一名成员每人支付了相同的金额，请一位嘉宾在数学俱乐部的会议上讲解解题方法。他们共付给这位嘉宾 。这个 位数中缺失的数字 是多少？
Eleven members of the Middle School Math Club each paid the same amount for a guest speaker to talk about problem solving at their math club meeting. They paid their guest speaker . What is the missing digit of this -digit number?
小提示：
名成员支付的总额必须能被 整除
The total paid by members must be divisible by .
大提示：
对三位数使用 的交错和整除判定
For a three-digit number, use the alternating-sum test for divisibility by .
解答：
因为 个人支付相同金额，所以总额能被 整除。因此 必须能被 整除。
的整除规则是：交错位数字和之差若能被 整除，则原数能被 整除。
对 ，这个差为 要让它能被 整除，只能有 。
所以正确答案是 D。
Note that since people paid the same amount, then the resulting sum is divisible by Therefore, must be divisible by
Remember the divisibility rule for : if we take the difference of the sums of alternating digits, and this difference is divisible by the whole number is divisible by
For the aforementioned difference is The only way for this to be divisible by is if
Thus, D is the correct answer.
其他年份的第 8 题
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