Question

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 be a factor of $Z?$

$11$

$19$

$101$

$111$

$1111$

Answer: A

Concepts:prime factorization place value

Difficulty rating: 940

Small Hint:

Let $n$ be the three-digit block that is repeated. Express $Z$ in terms of $n$ .

Big Hint:

Factor $1001$ .

Video solution:

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Written solution:

Let $n$ be the three-digit number formed by either repeated block. Then $\begin{aligned} Z&=1000n+n\\ &=1001n\\ &=7\cdot11\cdot13\cdot n. \end{aligned}$ Therefore, $11$ must be a factor of $Z.$

Thus, A is the correct answer.

2017 AMC 8 Problem 7