Difference between revisions of "2017 AMC 8 Problems/Problem 7"

Revision as of 06:47, 8 January 2023

Problem

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 are given one of the numbers that can represent $Z$ , so we can just try out the options to see which one is a factor of $247247$ . We get $\boxed{\textbf{(A)}\ 11}$ .

Solution 3

To find out when a number is divisible by 11, place plus and minus signs alternatively in front of every digit, then calculate the result. If this result is divisible by 11 (including 0), the number is divisible by 11; otherwise, the number isn’t divisible by 11. In this case, $+2-4+7-2+4-7=0$ . Because the result is 0, the number 247247 is divisible by 11 and so we get $\boxed{\textbf{(A)}\ 11}$ .

--LarryFlora

Solution 4

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 $\boxed{\textbf{(A)}\ 11}$ .

~PEKKA

Solution 5

We can find that all numbers like $Z$ are divisible by 1001. 1001 is divisible by 11 because when we divide it, we get a whole number. So, the answer is $\boxed{\textbf{(A)}\ 11}$ .

~AfterglowBlaziken

Video Solution by OmegaLearn

https://youtu.be/At4w8uylvv8?t=99 https://youtu.be/7an5wU9Q5hk?t=647

Video Solution

https://youtu.be/-saKu3lLU0U

~savannahsolver

@@ Line 24: / Line 24: @@
 ==Solution 5==
 We can find that all numbers like <math>Z</math> are divisible by 1001. 1001 is divisible by 11 because when we divide it, we get a whole number. So, the answer is <math>\boxed{\textbf{(A)}\ 11}</math>.
 ~AfterglowBlaziken

2017 AMC 8 (Problems • Answer Key • Resources)
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