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Difference between revisions of "2023 AMC 12B Problems/Problem 16"
m (→‎Solution 3 (mod 6))
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~JN
 
~JN
  
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==Solution 3 (mod 6)==
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==Solution 3 (Modulo 6)==
 +
Let the number of <math>6</math> cent coins be <math>a</math>, the number of <math>10</math> cent coins be <math>b</math>, and the number of <math>15</math> cent coins be <math>c</math>. We get the [https://artofproblemsolving.com/wiki/index.php/Diophantine_equation Diophantine equation]
 +
 
 +
<cmath>6a + 10b + 15c = k</cmath>
 +
 
 +
and we wish to find the largest possible value of <math>k</math>
 +
 
 +
Construct the following <math>\mod 6</math> table of <math>6</math>, <math>10</math>, and <math>15</math>
  
 
<cmath>\begin{array}{c|ccc}
 
<cmath>\begin{array}{c|ccc}
 
  &  &  & \\
 
  &  &  & \\
−
number of coins & 6 & 10 & 15 \\  
+
\text{number of coins} & 6 & 10 & 15 \\  
 
\hline
 
\hline
 
  &  &  & \\  
 
  &  &  & \\  
Line 90: Line 97:
  
 
  <math>0 \mod 6</math>
 
  <math>0 \mod 6</math>
−
  We can obtain any integer that is <math>0 \mod 6</math> because we have <math>6</math> cent coins.
+
  We can obtain any value that is <math>0 \mod 6</math> because we have <math>6</math> cent coins.
  
 
  <math>1 \mod 6</math>
 
  <math>1 \mod 6</math>
−
  We can obtain integers that equal <math>1 \mod 6</math> by using one <math>10</math> cent coin, one <math>15</math> cent coin, and as many <math>6</math> cent coins that are needed.
+
  We can obtain values that equal <math>1 \mod 6</math> by using one <math>10</math> cent coin, one <math>15</math> cent coin, and as many <math>6</math> cent coins needed.
−
  The smallest integer that equals <math>1 \mod 6</math> we can obtain is <math>10+15 = 25</math>. Therefore, the largest integer that equals <math>1 \mod 6</math> we cannot obtain is <math>25-6=19</math>
+
  The smallest value that equals <math>1 \mod 6</math> we can obtain is <math>10+15 = 25</math>. Therefore, the largest value that equals <math>1 \mod 6</math> we cannot obtain is <math>25-6=19</math>
  
 
  <math>2 \mod 6</math>
 
  <math>2 \mod 6</math>
−
  We can obtain integers that equal <math>2 \mod 6</math> by using two <math>10</math> cent coins, and as many <math>6</math> cent coins that are needed.
+
  We can obtain values that equal <math>2 \mod 6</math> by using two <math>10</math> cent coins, and as many <math>6</math> cent coins as needed.
−
  The smallest integer that equals <math>2 \mod 6</math> we can obtain is <math>10+10 = 20</math>. Therefore, the largest integer that equals <math>2 \mod 6</math> we cannot obtain is <math>20-6=14</math>
+
  The smallest value that equals <math>2 \mod 6</math> we can obtain is <math>10+10 = 20</math>. Therefore, the largest value that equals <math>2 \mod 6</math> we cannot obtain is <math>20-6=14</math>
  
 
  <math>3 \mod 6</math>
 
  <math>3 \mod 6</math>
−
  We can obtain integers that equal <math>3 \mod 6</math> by using one <math>15</math> cent coin, and as many <math>6</math> cent coins that are needed.
+
  We can obtain values that equal <math>3 \mod 6</math> by using one <math>15</math> cent coin, and as many <math>6</math> cent coins as needed.
−
  The smallest integer that equals <math>3 \mod 6</math> we can obtain is <math>15</math>. Therefore, the largest integer that equals <math>3 \mod 6</math> we cannot obtain is <math>15-6=9</math>
+
  The smallest value that equals <math>3 \mod 6</math> we can obtain is <math>15</math>. Therefore, the largest value that equals <math>3 \mod 6</math> we cannot obtain is <math>15-6=9</math>
  
 
  <math>4 \mod 6</math>
 
  <math>4 \mod 6</math>
−
  We can obtain integers that equal <math>4 \mod 6</math> by using one <math>10</math> cent coin, and as many <math>6</math> cent coins that are needed.
+
  We can obtain values that equal <math>4 \mod 6</math> by using one <math>10</math> cent coin, and as many <math>6</math> cent coins as needed.
−
  The smallest integer that equals <math>4 \mod 6</math> we can obtain is <math>10</math>. Therefore, the largest integer that equals <math>4 \mod 6</math> we cannot obtain is <math>10-6=4</math>
+
  The smallest value that equals <math>4 \mod 6</math> we can obtain is <math>10</math>. Therefore, the largest value that equals <math>4 \mod 6</math> we cannot obtain is <math>10-6=4</math>
  
 
  <math>5 \mod 6</math>
 
  <math>5 \mod 6</math>
−
  We can obtain integers that equal <math>5 \mod 6</math> by using two <math>10</math> cent coins, one <math>15</math> cent coin, and as many <math>6</math> cent coins that are needed.
+
  We can obtain values that equal <math>5 \mod 6</math> by using two <math>10</math> cent coins, one <math>15</math> cent coin, and as many <math>6</math> cent coins as needed.
−
  The smallest integer that equals <math>5 \mod 6</math> we can obtain is <math>10+10+15 = 35</math>. Therefore, the largest integer that equals <math>5 \mod 6</math> we cannot obtain is <math>35-6=29</math>
+
  The smallest value that equals <math>5 \mod 6</math> we can obtain is <math>10+10+15 = 35</math>. Therefore, the largest value that equals <math>5 \mod 6</math> we cannot obtain is <math>35-6=29</math>
  
−
Hence, the largest integer we cannot obtain is <math>29</math>, <math>2 + 9 = \boxed{\textbf{(D) 11}}</math>.
+
Hence, the largest value in cents we cannot obtain using <math>6</math>, <math>10</math>, and <math>15</math> cent coins is <math>29</math>, <math>2 + 9 = \boxed{\textbf{(D) 11}}</math>.
  
 
~[https://artofproblemsolving.com/wiki/index.php/User:Isabelchen isabelchen]
 
~[https://artofproblemsolving.com/wiki/index.php/User:Isabelchen isabelchen]

Revision as of 05:49, 16 November 2023

Problem

In the state of Coinland, coins have values $6,10,$ and $15$ cents. Suppose $x$ is the value in cents of the most expensive item in Coinland that cannot be purchased using these coins with exact change. What is the sum of the digits of $x?$

$\textbf{(A) }8\qquad\textbf{(B) }10\qquad\textbf{(C) }7\qquad\textbf{(D) }11\qquad\textbf{(E) }9$

Solution 1

This problem asks to find largest $x$ that cannot be written as \[ 6 a + 10 b + 15 c = x, \hspace{1cm} (1) \] where $a, b, c \in \Bbb Z_+$.

Denote by $r \in \left\{ 0, 1 \right\}$ the remainder of $x$ divided by 2. Modulo 2 on Equation (1), we get By using modulus $m \in \left\{ 2, 3, 5 \right\}$ on the equation above, we get $c \equiv r \pmod{2}$.

Following from Chicken McNugget's theorem, we have that any number that is no less than $(3-1)(5-1) = 8$ can be expressed in the form of $3a + 5b$ with $a, b \in \Bbb Z_+$.

Therefore, all even numbers that are at least equal to $2 \cdot 8 + 15 \cdot 0 = 16$ can be written in the form of Equation (1) with $a, b, c \in \Bbb Z_+$. All odd numbers that are at least equal to $2 \cdot 8 + 15 \cdot 1 = 31$ can be written in the form of Equation (1) with $a, b, c \in \Bbb Z_+$.

The above two cases jointly imply that all numbers that are at least 30 can be written in the form of Equation (1) with $a, b, c \in \Bbb Z_+$.

Next, we need to prove that 29 cannot be written in the form of Equation (1) with $a, b, c \in \Bbb Z_+$.

Because 29 is odd, we must have $c \equiv 1 \pmod{2}$. Because $a, b, c \in \Bbb Z_+$, we must have $c = 1$. Plugging this into Equation (1), we get $3 a + 5 b = 7$. However, this equation does not have non-negative integer solutions.

All analysis above jointly imply that the largest $x$ that has no non-negative integer solution to Equation (1) is 29. Therefore, the answer is $2 + 9 = \boxed{\textbf{(D) 11}}$.

~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)

Solution 2

Arrange the positive integers into rows of 6, like this: \begin{align*} 01, 02, 03, 04, 05, 06 \\ 07, 08, 09, 10, 11, 12 \\ 13, 14, 15, 16, 17, 18 \\ 19, 20, 21, 22, 23, 24 \\ 25, 26, 27, 28, 29, 30 \\ 31, 32, 33, 34, 35, 36 \\ ... \end{align*}

Observe that if any number can be made from a combination of $6$s, $10$s, and $15$s, then every number below it in the same column must also be possible to make, by simply adding 6s.

Thus, we will cross out any numbers that CAN be made as well as all numbers below it.

In column 1, 25 is possible (10+15) and so is everything below 25.

Column 2 - cross out 20 (10+10) and everything below it

Column 3 - cross out 15 and everything below it

Column 4 - cross out 10 and everything below it

Column 5 - cross out 35 (15+10+10) and everything below it

Column 6 - all numbers here are possible, so cross all out.

The maximum number that remains is 29. Answer is 2+9=11.

(sorry for the bad formatting - feel free to edit)

~JN

Solution 3 (Modulo 6)

Let the number of $6$ cent coins be $a$, the number of $10$ cent coins be $b$, and the number of $15$ cent coins be $c$. We get the Diophantine equation

\[6a + 10b + 15c = k\]

and we wish to find the largest possible value of $k$

Construct the following $\mod 6$ table of $6$, $10$, and $15$

\[\begin{array}{c|ccc} & & & \\ \text{number of coins} & 6 & 10 & 15 \\ \hline & & & \\ 1 & 0 & 4 & 3\\ & & & \\ 2 & 0 & 2 & 0 \\ \end{array}\]

There are only $6$ possible residues for $6$, they are: $0$, $1$, $2$, $3$, $4$, and $5$. 

$0 \mod 6$
We can obtain any value that is $0 \mod 6$ because we have $6$ cent coins.

$1 \mod 6$
We can obtain values that equal $1 \mod 6$ by using one $10$ cent coin, one $15$ cent coin, and as many $6$ cent coins needed.
The smallest value that equals $1 \mod 6$ we can obtain is $10+15 = 25$. Therefore, the largest value that equals $1 \mod 6$ we cannot obtain is $25-6=19$

$2 \mod 6$
We can obtain values that equal $2 \mod 6$ by using two $10$ cent coins, and as many $6$ cent coins as needed.
The smallest value that equals $2 \mod 6$ we can obtain is $10+10 = 20$. Therefore, the largest value that equals $2 \mod 6$ we cannot obtain is $20-6=14$

$3 \mod 6$
We can obtain values that equal $3 \mod 6$ by using one $15$ cent coin, and as many $6$ cent coins as needed.
The smallest value that equals $3 \mod 6$ we can obtain is $15$. Therefore, the largest value that equals $3 \mod 6$ we cannot obtain is $15-6=9$

$4 \mod 6$
We can obtain values that equal $4 \mod 6$ by using one $10$ cent coin, and as many $6$ cent coins as needed.
The smallest value that equals $4 \mod 6$ we can obtain is $10$. Therefore, the largest value that equals $4 \mod 6$ we cannot obtain is $10-6=4$

$5 \mod 6$
We can obtain values that equal $5 \mod 6$ by using two $10$ cent coins, one $15$ cent coin, and as many $6$ cent coins as needed.
The smallest value that equals $5 \mod 6$ we can obtain is $10+10+15 = 35$. Therefore, the largest value that equals $5 \mod 6$ we cannot obtain is $35-6=29$

Hence, the largest value in cents we cannot obtain using $6$, $10$, and $15$ cent coins is $29$, $2 + 9 = \boxed{\textbf{(D) 11}}$.

~isabelchen

Video Solution 1 by OmegaLearn

https://youtu.be/E8WsGfn95mk


See Also

2023 AMC 12B (Problems • Answer Key • Resources)
Preceded by
Problem 15 		Followed by
Problem 17
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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