2018 AMC 8 第 7 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。
7.
位数 能被 整除。这个数除以 的余数是多少？
The -digit number is divisible by What is the remainder when this number is divided by
小提示：
先用 的整除规则确定
Use the divisibility-by- rule to determine .
大提示：
确定后，求模 的余数只需要看最后三位
Once is known, only the last three digits matter for the remainder modulo .
视频讲解：
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文字解答：
注意，一个数能被 整除，当且仅当它的各位数字之和也能被 整除。
题中这个 位数的数字和为 因为 能被 整除， 也必须被 整除。又因为 是一位数字，，所以 只能是 。
现在可知这个 位数是 ，而我们要求 除以 的余数。长除法给出 ，所以余数是 。
所以正确答案是 B。
Notice that a number is divisible by if and only if the sum of its digits is also divisible by
The sum of the digits of the -digit number in the problem is: As is divisible by must also be divisible by Also, as is a digit, we know that This means that can only be
Now we know that the -digit number in question is and we want to find the remainder when we divide by To solve this, simply use long division to see that Therefore, the remainder is
Thus, the correct answer is B.
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