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

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==Problem 7==
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==Problem==
  
 
Let <math>Z</math> 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 <math>Z</math>?
 
Let <math>Z</math> 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 <math>Z</math>?
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<math>\textbf{(A) }11\qquad\textbf{(B) }19\qquad\textbf{(C) }101\qquad\textbf{(D) }111\qquad\textbf{(E) }1111</math>
 
<math>\textbf{(A) }11\qquad\textbf{(B) }19\qquad\textbf{(C) }101\qquad\textbf{(D) }111\qquad\textbf{(E) }1111</math>
  
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==Solution 1==
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==Solution 1 (detailed explanation)==
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To check, if a number is divisible by 19, take its unit digit and multiply it by 2, then add the result to the rest of the number, and repeat this step until the number is reduced to two digits. If the result is divisible by 19, then the original number is also divisible by 19. Or we could just try to divide the example number the problem gave us by 19.
  
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Let <math>Z = \overline{ABCABC} = 1001 \cdot \overline{ABC} = 7 \cdot 11 \cdot 13 \cdot \overline{ABC}.</math> Clearly, <math>Z</math> is divisible by <math>\boxed{\textbf{(A)}\ 11}</math>.
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After we got 19 eliminated, we can see that the other options have a lot of 1's in them. The divisibility rule for 11 is add alternating digits up, then take the difference of them. The example number works like that. If we add variables, ABCDEF to make number ABCABC, we can see that (A+C+B) - (B+A+C) = 0. Which is divisible by 11, so our answer choice is <math>\boxed{\textbf{(A)}\ 11}</math>.
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by: CHECKMATE2021 (edited by CHECKMATE2021)
  
 
==Solution 2==
 
==Solution 2==
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We are given one of the numbers that can represent <math>Z</math>, so we can just try out the options to see which one is a factor of <math>247247</math>. We get <math>\boxed{\textbf{(A)}\ 11}</math>.
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~CHECKMATE2021
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==Solution 3( similar to solution 1)==
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Similar to solution 1, let <math>Z=ABCABC</math>. To prove it is divisible by 11, we can compute its alternating sum, which is <math>A-B+C-A+B-C=0</math>, which is divisible by 11. Therefore, the answer is <math>\boxed{\textbf{(A)}\ 11}</math>.
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~PEKKA
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==Solution 4==
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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>.
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 +
~AfterglowBlaziken
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 +
==Video Solution (CREATIVE THINKING!!!)==
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https://youtu.be/WVHPSVOXE4I
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~Education, the Study of Everything
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==Video Solution by OmegaLearn==
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https://youtu.be/At4w8uylvv8?t=99
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https://youtu.be/7an5wU9Q5hk?t=647
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==Video Solution==
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https://youtu.be/-saKu3lLU0U
  
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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. -Baolan (Hi MVMS)
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~savannahsolver
  
 
==See Also==
 
==See Also==

Latest revision as of 19:43, 2 July 2024

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 (detailed explanation)

To check, if a number is divisible by 19, take its unit digit and multiply it by 2, then add the result to the rest of the number, and repeat this step until the number is reduced to two digits. If the result is divisible by 19, then the original number is also divisible by 19. Or we could just try to divide the example number the problem gave us by 19.

After we got 19 eliminated, we can see that the other options have a lot of 1's in them. The divisibility rule for 11 is add alternating digits up, then take the difference of them. The example number works like that. If we add variables, ABCDEF to make number ABCABC, we can see that (A+C+B) - (B+A+C) = 0. Which is divisible by 11, so our answer choice is $\boxed{\textbf{(A)}\ 11}$.

by: CHECKMATE2021 (edited by CHECKMATE2021)

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

~CHECKMATE2021

Solution 3( similar to solution 1)

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 4

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 (CREATIVE THINKING!!!)

https://youtu.be/WVHPSVOXE4I

~Education, the Study of Everything

Video Solution by OmegaLearn

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

Video Solution

https://youtu.be/-saKu3lLU0U

~savannahsolver

See Also

2017 AMC 8 (Problems • Answer Key • Resources)
Preceded by
Problem 6 		Followed by
Problem 8
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AJHSME/AMC 8 Problems and Solutions

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