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Difference between revisions of "2013 AMC 12A Problems/Problem 17"

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==Solution==
 
==Solution==
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===Solution 1===
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The first pirate takes <math>\frac{1}{12}</math> of the <math>x</math> coins, leaving <math>\frac{11}{12} x</math>.
 +
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The second pirate takes <math>\frac{2}{12}</math> of the remaining coins, leaving <math>\frac{10}{12}\cdot
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\frac{11}{12}*x</math>.
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 +
Note that
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 +
<math>12^{11} = (2^2 \cdot 3)^{11} = 2^{22} \cdot 3^{11}</math>
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 +
<math>11! = 11 \cdot 10 \cdot 9 \cdot 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2</math>
  
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The first pirate takes <math>\frac{1}{12}</math> of the <math>x</math> coins, leaving <math>\frac{11}{12} x</math>.
 
  
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The second pirate takes <math>\frac{2}{12}</math> of the remaining coins, leaving <math>\frac{10}{12}*\frac{11}{12}*x</math>.
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All the <math>2</math>s and <math>3</math>s cancel out of <math>11!</math>, leaving
  
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Continuing this pattern, the eleventh pirate must take <math>\frac{11}{12}</math> of the remaining coins after the first ten pirates have taken their share, which leaves <math>\frac{11!}{12^{11}}*x</math>. The twelfth pirate takes all of this.
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<math>11 \cdot 5 \cdot 7 \cdot 5 = 1925</math>
  
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in the numerator.
  
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Note that
 
  
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<math>12^{12} = (2^2 * 3)^{11} = 2^{22} * 3^{11}</math>
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We know there were just enough coins to cancel out the denominator in the fraction. So, at minimum, <math>x</math> is the denominator, leaving <math>\boxed{\textbf{(D) }1925}</math> coins for the twelfth pirate.
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===Solution 2===
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The answer cannot be an even number. Here is why:
  
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<math>11! = 11 * 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2</math>
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Consider the highest power of 2 that divides the starting number of coins, and consider how this value changes as each pirate takes their share. At each step, the size of the pile is multiplied by some <math>\frac{n}{12}</math>. This means the highest power of 2 that divides the number of coins is continually decreasing or staying the same (except once, briefly, when we multiply by <math>\frac{2}{3}</math> for pirate 4, but it immediately drops again in the next step).
  
 +
Therefore, if the 12th pirate's coin total were even, then it can't be the smallest possible value, because we can safely cut the initial pot (and all the intermediate totals) in half. We could continue halving the result until the 12th pirate's total is finally odd.
  
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All the 2s and 3s cancel out of <math>11!</math>, leaving
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Only one of the choices given is odd, <math>\boxed{\textbf{(D) }1925}</math>.
  
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<math>11 * 5 * 7 * 5 = 1925</math>
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Note that if there had been multiple odd answer choices, we could repeat the reasoning above to see that the value also must not be divisible by 3.
  
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in the numerator.
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==Solution 3 (Work backwards)==
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Let <math>x</math> be the number of coins the <math>12</math>th pirate takes. Then the number of coins the <math>k<12</math>th pirate takes is <math>\frac{12}{1} \cdot \frac{12}{2} \cdots \frac{12}{12-k} x</math>. For all these to be an integer, we need the denominators to divide into the numerators. Looking at prime factors, obviously there are sufficient <math>2</math>s and <math>3</math>s, so we just need <math>5^2 \cdot 7 \cdot 11 = \boxed{\textbf{(D) }1925}</math> to divide into <math>x</math>. -Frestho
  
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==Video Solution by Richard Rusczyk==
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https://artofproblemsolving.com/videos/amc/2013amc12a/356
  
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We know there were just enough coins to cancel out the denominator in the fraction. So, at minimum, <math>x</math> is the denominator, leaving <math>1925</math> coins for the twelfth pirate.
+
~dolphin7
  
 
== See also ==
 
== See also ==
 
{{AMC12 box|year=2013|ab=A|num-b=16|num-a=18}}
 
{{AMC12 box|year=2013|ab=A|num-b=16|num-a=18}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 00:36, 11 October 2020

Problem 17

A group of $12$ pirates agree to divide a treasure chest of gold coins among themselves as follows. The $k^\text{th}$ pirate to take a share takes $\frac{k}{12}$ of the coins that remain in the chest. The number of coins initially in the chest is the smallest number for which this arrangement will allow each pirate to receive a positive whole number of coins. How many coins does the $12^{\text{th}}$ pirate receive?

$\textbf{(A)} \ 720 \qquad \textbf{(B)} \ 1296 \qquad \textbf{(C)} \ 1728 \qquad \textbf{(D)} \ 1925 \qquad \textbf{(E)} \ 3850$

Solution

Solution 1

The first pirate takes $\frac{1}{12}$ of the $x$ coins, leaving $\frac{11}{12} x$.

The second pirate takes $\frac{2}{12}$ of the remaining coins, leaving $\frac{10}{12}\cdot \frac{11}{12}*x$.

Note that

$12^{11} = (2^2 \cdot 3)^{11} = 2^{22} \cdot 3^{11}$

$11! = 11 \cdot 10 \cdot 9 \cdot 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2$


All the $2$s and $3$s cancel out of $11!$, leaving

$11 \cdot 5 \cdot 7 \cdot 5 = 1925$

in the numerator.


We know there were just enough coins to cancel out the denominator in the fraction. So, at minimum, $x$ is the denominator, leaving $\boxed{\textbf{(D) }1925}$ coins for the twelfth pirate.

Solution 2

The answer cannot be an even number. Here is why:

Consider the highest power of 2 that divides the starting number of coins, and consider how this value changes as each pirate takes their share. At each step, the size of the pile is multiplied by some $\frac{n}{12}$. This means the highest power of 2 that divides the number of coins is continually decreasing or staying the same (except once, briefly, when we multiply by $\frac{2}{3}$ for pirate 4, but it immediately drops again in the next step).

Therefore, if the 12th pirate's coin total were even, then it can't be the smallest possible value, because we can safely cut the initial pot (and all the intermediate totals) in half. We could continue halving the result until the 12th pirate's total is finally odd.

Only one of the choices given is odd, $\boxed{\textbf{(D) }1925}$.

Note that if there had been multiple odd answer choices, we could repeat the reasoning above to see that the value also must not be divisible by 3.

Solution 3 (Work backwards)

Let $x$ be the number of coins the $12$th pirate takes. Then the number of coins the $k<12$th pirate takes is $\frac{12}{1} \cdot \frac{12}{2} \cdots \frac{12}{12-k} x$. For all these to be an integer, we need the denominators to divide into the numerators. Looking at prime factors, obviously there are sufficient $2$s and $3$s, so we just need $5^2 \cdot 7 \cdot 11 = \boxed{\textbf{(D) }1925}$ to divide into $x$. -Frestho

Video Solution by Richard Rusczyk

https://artofproblemsolving.com/videos/amc/2013amc12a/356

~dolphin7

See also

2013 AMC 12A (Problems • Answer Key • Resources)
Preceded by
Problem 16 		Followed by
Problem 18
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 AMC 12 Problems and Solutions

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