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Difference between revisions of "2018 AMC 8 Problems/Problem 3"

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==Problem 3==
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==Problem==
 
Students Arn, Bob, Cyd, Dan, Eve, and Fon are arranged in that order in a circle. They start counting: Arn first, then Bob, and so forth. When the number contains a 7 as a digit (such as 47) or is a multiple of 7 that person leaves the circle and the counting continues. Who is the last one present in the circle?
 
Students Arn, Bob, Cyd, Dan, Eve, and Fon are arranged in that order in a circle. They start counting: Arn first, then Bob, and so forth. When the number contains a 7 as a digit (such as 47) or is a multiple of 7 that person leaves the circle and the counting continues. Who is the last one present in the circle?
  
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<math>\textbf{(A) } \text{Arn}\qquad\textbf{(B) }\text{Bob}\qquad\textbf{(C) }\text{Cyd}\qquad\textbf{(D) }\text{Dan}\qquad \textbf{(E) }\text{Eve}\qquad</math>
  
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==Solution==
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==Solution 1(Simulation)==
 
The five numbers which cause people to leave the circle are <math>7, 14, 17, 21,</math> and <math>27.</math>
 
The five numbers which cause people to leave the circle are <math>7, 14, 17, 21,</math> and <math>27.</math>
  
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Arn counts <math>7</math> (assuming they start at <math>1</math>) so he leaves first. Then Cyd counts <math>14</math>, as there are <math>7</math> numbers to be counted from this point. Then Fon, Bob, Eve, so last one standing is Dan, <math>\textbf{(D)}</math> -scrabbler94
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The most straightforward way to do this would be to draw out the circle with the people, and cross off people as you count.
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Assuming the six people start with <math>1</math>, Arn counts <math>7</math> so he leaves first. Then, Cyd counts <math>14</math> as there are <math>7</math> numbers to be counted from this point. Then, Fon, Bob, and Eve, count, <math>17,</math> <math>21,</math> and <math>27</math>, respectively, so the last one standing is Dan. Hence, the answer would be <math>\boxed{\textbf{(D) }\text{Dan}}</math>.
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~Nivaar
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==Solution 2==
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Assign each person a position: Arn is in position <math>1</math>, Bob in <math>2</math>, etc. Note that if there are <math>n</math> people standing in a circle, a person will say a number that is congruent to their position modulo <math>n</math>. It follows that if there are <math>k</math> numbers to be said (with someone leaving on the <math>kth</math>), the person that leaves will stand in position <math>k\pmod{n}.</math>
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The five numbers which cause people to leave the circle are <math>7, 14, 17, 21,</math> and <math>27.</math> Since there are initially <math>6</math> people in the circle, so the person who leaves stands in position <math>7\equiv 1\pmod 6</math>, or position <math>1</math>.
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Arn now leaves, and because Bob is the next person to say a number, we reassign the positions such that Bob is in position <math>1</math>, Cyd in <math>2</math>, and Fon in <math>5</math>. There are <math>7</math> more numbers to be said, and someone leaves on the <math>7th</math>. It follows that the person who leaves stands in position <math>7\equiv 2\pmod 5</math>.
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Cyd leaves the circle, and <math>4</math> people remain, namely Bob, Dan, Eve, and Fon. Assign the position numbers such that Dan has <math>1</math>, Eve <math>2</math>, etc. There are <math>3</math> more numbers to be said, and someone leaves on the <math>3rd</math>. The person standing in position <math>3</math> (Fon) now leaves.
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There are two people left, Eve and Dan, with Dan in position <math>1</math> now. There are <math>6</math> more numbers to be said, and Eve is standing in position <math>6\equiv 0\pmod 2</math>, so she leaves.
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The last person standing is <math>\boxed{\textbf{(D) }\text{Dan}}</math>.
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-Benedict T (countmath1)
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== Video Solution (CRITICAL THINKING!!!)==
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https://youtu.be/wziaWxZmjgg
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~Education, the Study of Everything
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==Video Solution==
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https://youtu.be/uK4bbVpwHGc
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~savannahsolver
  
 
==See Also==
 
==See Also==

Latest revision as of 12:10, 23 January 2024

Problem

Students Arn, Bob, Cyd, Dan, Eve, and Fon are arranged in that order in a circle. They start counting: Arn first, then Bob, and so forth. When the number contains a 7 as a digit (such as 47) or is a multiple of 7 that person leaves the circle and the counting continues. Who is the last one present in the circle?

$\textbf{(A) } \text{Arn}\qquad\textbf{(B) }\text{Bob}\qquad\textbf{(C) }\text{Cyd}\qquad\textbf{(D) }\text{Dan}\qquad \textbf{(E) }\text{Eve}\qquad$

Solution 1(Simulation)

The five numbers which cause people to leave the circle are $7, 14, 17, 21,$ and $27.$

The most straightforward way to do this would be to draw out the circle with the people, and cross off people as you count.

Assuming the six people start with $1$, Arn counts $7$ so he leaves first. Then, Cyd counts $14$ as there are $7$ numbers to be counted from this point. Then, Fon, Bob, and Eve, count, $17,$ $21,$ and $27$, respectively, so the last one standing is Dan. Hence, the answer would be $\boxed{\textbf{(D) }\text{Dan}}$.

~Nivaar

Solution 2

Assign each person a position: Arn is in position $1$, Bob in $2$, etc. Note that if there are $n$ people standing in a circle, a person will say a number that is congruent to their position modulo $n$. It follows that if there are $k$ numbers to be said (with someone leaving on the $kth$), the person that leaves will stand in position $k\pmod{n}.$

The five numbers which cause people to leave the circle are $7, 14, 17, 21,$ and $27.$ Since there are initially $6$ people in the circle, so the person who leaves stands in position $7\equiv 1\pmod 6$, or position $1$.

Arn now leaves, and because Bob is the next person to say a number, we reassign the positions such that Bob is in position $1$, Cyd in $2$, and Fon in $5$. There are $7$ more numbers to be said, and someone leaves on the $7th$. It follows that the person who leaves stands in position $7\equiv 2\pmod 5$.

Cyd leaves the circle, and $4$ people remain, namely Bob, Dan, Eve, and Fon. Assign the position numbers such that Dan has $1$, Eve $2$, etc. There are $3$ more numbers to be said, and someone leaves on the $3rd$. The person standing in position $3$ (Fon) now leaves.

There are two people left, Eve and Dan, with Dan in position $1$ now. There are $6$ more numbers to be said, and Eve is standing in position $6\equiv 0\pmod 2$, so she leaves.

The last person standing is $\boxed{\textbf{(D) }\text{Dan}}$.

-Benedict T (countmath1)

Video Solution (CRITICAL THINKING!!!)

https://youtu.be/wziaWxZmjgg

~Education, the Study of Everything

Video Solution

https://youtu.be/uK4bbVpwHGc

~savannahsolver

See Also

2018 AMC 8 (Problems • Answer Key • Resources)
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Problem 2 		Followed by
Problem 4
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All AJHSME/AMC 8 Problems and Solutions

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