2017 AMC 8 Problems/Problem 7

Problem 7

Let $Z$ 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 $Z$?

$\textbf{(A) }11\qquad\textbf{(B) }19\qquad\textbf{(C) }101\qquad\textbf{(D) }111\qquad\textbf{(E) }1111$

Solution 1

Let $Z = \overline{ABCABC} = 1001 \cdot \overline{ABC} = 7 \cdot 11 \cdot 13 \cdot \overline{ABC}.$ Clearly, $Z$ is divisible by $\boxed{\textbf{(A)}\ 11}$.

Solution 2

We can see that numbers like $247247$ can be written as $ABCABC$. We can see that the alternating sum of digits is $C-B+A-C+B-A$, which is $0$. Because $0$ is a multiple of $11$, any number $ABCABC$ is a multiple of $11$, so the answer is $\boxed{\textbf{(A)}\ 11}$.

See Also

2017 AMC 8 (ProblemsAnswer KeyResources)
Preceded by
Problem 6
Followed by
Problem 8
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All AJHSME/AMC 8 Problems and Solutions

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