AMC8 2017
AMC8 2017 · Q7
AMC8 2017 · Q7. It mainly tests Digit properties (sum of digits, divisibility tests).
Let Z be a 6-digit positive integer, such as 123,123, whose first three digits are the same as its last three digits taken in the same order. Which of the following numbers must be a factor of Z?
设 Z 是一个 6 位正整数，例如 123123，其前三位数字与后三位数字顺序相同。以下哪个数一定是 Z 的因数？
(A)
11
11
(B)
19
19
(C)
101
101
(D)
111
111
(E)
1111
1111
Answer
Correct choice: (A)
正确答案：（A）
Solution
Answer (A): Assume Z has the form abcabc. Then
\[ Z = 1001 \cdot abc = 7 \cdot 11 \cdot 13 \cdot abc \]
So 11 must be a factor of Z.
答案（A）：假设 $Z$ 的形式为 abcabc。于是
\[ Z = 1001 \cdot abc = 7 \cdot 11 \cdot 13 \cdot abc \]
所以 11 必须是 $Z$ 的因数。
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