AMC 8 2017 Problem 7
AMC8 2017 年第 7 题
整除与因数★★☆☆☆
Let be a 6-digit positive integer, such as 247247, whose first three digits are the same as its last three digits taken in the same order. Which of the following numbers must also be a factor of ?
Z 表示一个 6 位的正整数，例如 247247，从左往右前三位数字和后三位的数字一样。下面哪个数一定也是 Z 的一个因子？
- A.11
- B.19
- C.101
- D.111
- E.1111
正确答案 · Correct Answer
A
解析 · Solution
Solution 1
We are given one of the numbers that can represent , so we can just try out the options to see which one is a factor of 247247. By the previous solution, 19 does not work even though it is a factor of the example number. We get 11. Similar to solution 1, let Z=ABCABC. To prove it is divisible by 11, we can compute its alternating sum, which is A-B+C-A+B-C=0, which is divisible by 11. Therefore, the answer is 11.
We can find that all numbers like are divisible by 1001. 1001 is 7 x 11 x 13 and when we divide 11, we get a whole number. So, the answer is 11. by Geometry1421
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