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Difference between revisions of "2017 AMC 10B Problems/Problem 1"

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<math>\textbf{(A)}\ 11\qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 13\qquad\textbf{(D)}\ 14\qquad\textbf{(E)}\ 15</math>
 
<math>\textbf{(A)}\ 11\qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 13\qquad\textbf{(D)}\ 14\qquad\textbf{(E)}\ 15</math>
  
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==Solution==
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==Solution 1==
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Let her <math>2</math>-digit number be <math>x</math>. Multiplying by <math>3</math> makes it a multiple of <math>3</math>, meaning that the sum of its digits is divisible by <math>3</math>. Adding on <math>11</math> increases the sum of the digits by <math>1+1 = 2,</math> (we can ignore numbers such as <math>39+11=50</math>) and reversing the digits keeps the sum of the digits the same; this means that the resulting number must be <math>2</math> more than a multiple of <math>3</math>. There are two such numbers between <math>71</math> and <math>75</math>: <math>71</math> and <math>74.</math> Now that we have narrowed down the choices, we can simply test the answers to see which one will provide a two-digit number when the steps are reversed:
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===Solution 1===
 
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Let her <math>2</math>-digit number be <math>x</math>. Multiplying by <math>3</math> makes it a multiple of <math>3</math>, meaning that the sum of its digits is divisible by <math>3</math>. Adding on <math>11</math> increases the sum of the digits by <math>1+1 = 2,</math> (we can ignore that numbers such as <math>39+11=50</math> since the numbers must be between <math>71</math> and <math>75</math>) and reversing the digits keeps the sum of the digits the same; this means that the resulting number must be <math>2</math> more than a multiple of <math>3</math>. There are two such numbers between <math>71</math> and <math>75</math>: <math>71</math> and <math>74.</math> Now that we have narrowed down the choices, we can simply test the answers to see which one will provide a two-digit number when the steps are reversed:
 
 
<cmath></cmath>
 
<cmath></cmath>
 
For <math>71,</math> we reverse the digits, resulting in <math>17.</math> Subtracting <math>11</math>, we get <math>6.</math> We can already see that dividing this by <math>3</math> will not be a two-digit number, so <math>71</math> does not meet our requirements.
 
For <math>71,</math> we reverse the digits, resulting in <math>17.</math> Subtracting <math>11</math>, we get <math>6.</math> We can already see that dividing this by <math>3</math> will not be a two-digit number, so <math>71</math> does not meet our requirements.
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Therefore, our answer is <math>\boxed{\bold{(B)} 12}</math>.
 
Therefore, our answer is <math>\boxed{\bold{(B)} 12}</math>.
  
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===Solution 2===
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==Solution 2==
 
Working backwards, we reverse the digits of each number from <math>71</math>~<math>75</math> and subtract <math>11</math> from each, so we have
 
Working backwards, we reverse the digits of each number from <math>71</math>~<math>75</math> and subtract <math>11</math> from each, so we have
 
<cmath>6, 16, 26, 36, 46</cmath>
 
<cmath>6, 16, 26, 36, 46</cmath>
 
The only numbers from this list that are divisible by <math>3</math> are <math>6</math> and <math>36</math>. We divide both by <math>3</math>, yielding <math>2</math> and <math>12</math>. Since <math>2</math> is not a two-digit number, the answer is <math>\boxed{\textbf{(B)}\ 12}</math>.
 
The only numbers from this list that are divisible by <math>3</math> are <math>6</math> and <math>36</math>. We divide both by <math>3</math>, yielding <math>2</math> and <math>12</math>. Since <math>2</math> is not a two-digit number, the answer is <math>\boxed{\textbf{(B)}\ 12}</math>.
  
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===Solution 3===
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==Solution 3==
 
You can just plug in the numbers to see which one works. When you get to <math>12</math>, you multiply by <math>3</math> and add <math>11</math> to get <math>47</math>. When you reverse the digits of <math>47</math>, you get <math>74</math>, which is within the given range. Thus, the answer is  <math>\boxed{\textbf{(B)}\ 12}</math>.
 
You can just plug in the numbers to see which one works. When you get to <math>12</math>, you multiply by <math>3</math> and add <math>11</math> to get <math>47</math>. When you reverse the digits of <math>47</math>, you get <math>74</math>, which is within the given range. Thus, the answer is  <math>\boxed{\textbf{(B)}\ 12}</math>.
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==Solution 4 (Fastest Way)==
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Let x be the original number. The last digit of <math>3x+11</math> must be <math>7</math> so the last digit of <math>3x</math> must be <math>6</math>. The only answer choice that satisfies this is <math>\boxed{\textbf{(B)}\ 12}</math>.
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==Solution 5==
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Subtract <math>11</math> from the numbers <math>71</math> through <math>75</math>. This yields <math>71-11 = 60</math>, <math>72-11 = 61</math>, <math>73-11 = 62</math>, <math>74-11 = 63</math>, and <math>75-11 = 64</math>. Of these, the only ones divisible by <math>3</math> are <math>60</math> and <math>63</math>. Therefore, the only possible values are <math>71</math> and <math>74</math>. Switching the digits of each, we get <math>17</math> and <math>47</math>. Subtracting <math>11</math> from each, we get the numbers <math>6</math> and <math>36</math>. Dividing each by <math>3</math>, we get <math>2</math> and <math>12</math>. The only two-digit number is <math>12</math>, so the answer is <math>\boxed{\textbf{(B)}\ 12}</math>.
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~TheGoldenRetriever
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 +
==Video Solution (HOW TO CRITICALLY THINK!!!)==
 +
https://youtu.be/EbFw47vZASs
 +
 +
~Education, the Study of Everything
 +
 +
 +
 +
==Video Solution==
 +
https://youtu.be/PQnA07go4GM
 +
 +
==Video Solution ==
 +
https://youtu.be/zTGuz6EoBWY
  
 
==See Also==
 
==See Also==
 
{{AMC10 box|year=2017|ab=B|before=First Problem|num-a=2}}
 
{{AMC10 box|year=2017|ab=B|before=First Problem|num-a=2}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 16:51, 20 August 2023

Problem

Mary thought of a positive two-digit number. She multiplied it by $3$ and added $11$. Then she switched the digits of the result, obtaining a number between $71$ and $75$, inclusive. What was Mary's number?

$\textbf{(A)}\ 11\qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 13\qquad\textbf{(D)}\ 14\qquad\textbf{(E)}\ 15$

Solution 1

Let her $2$-digit number be $x$. Multiplying by $3$ makes it a multiple of $3$, meaning that the sum of its digits is divisible by $3$. Adding on $11$ increases the sum of the digits by $1+1 = 2,$ (we can ignore numbers such as $39+11=50$) and reversing the digits keeps the sum of the digits the same; this means that the resulting number must be $2$ more than a multiple of $3$. There are two such numbers between $71$ and $75$: $71$ and $74.$ Now that we have narrowed down the choices, we can simply test the answers to see which one will provide a two-digit number when the steps are reversed: \[\] For $71,$ we reverse the digits, resulting in $17.$ Subtracting $11$, we get $6.$ We can already see that dividing this by $3$ will not be a two-digit number, so $71$ does not meet our requirements. \[\] Therefore, the answer must be the reversed steps applied to $74.$ We have the following: \[\] $74\rightarrow47\rightarrow36\rightarrow12$ \[\] Therefore, our answer is $\boxed{\bold{(B)} 12}$.

Solution 2

Working backwards, we reverse the digits of each number from $71$~$75$ and subtract $11$ from each, so we have \[6, 16, 26, 36, 46\] The only numbers from this list that are divisible by $3$ are $6$ and $36$. We divide both by $3$, yielding $2$ and $12$. Since $2$ is not a two-digit number, the answer is $\boxed{\textbf{(B)}\ 12}$.

Solution 3

You can just plug in the numbers to see which one works. When you get to $12$, you multiply by $3$ and add $11$ to get $47$. When you reverse the digits of $47$, you get $74$, which is within the given range. Thus, the answer is $\boxed{\textbf{(B)}\ 12}$.


Solution 4 (Fastest Way)

Let x be the original number. The last digit of $3x+11$ must be $7$ so the last digit of $3x$ must be $6$. The only answer choice that satisfies this is $\boxed{\textbf{(B)}\ 12}$.

Solution 5

Subtract $11$ from the numbers $71$ through $75$. This yields $71-11 = 60$, $72-11 = 61$, $73-11 = 62$, $74-11 = 63$, and $75-11 = 64$. Of these, the only ones divisible by $3$ are $60$ and $63$. Therefore, the only possible values are $71$ and $74$. Switching the digits of each, we get $17$ and $47$. Subtracting $11$ from each, we get the numbers $6$ and $36$. Dividing each by $3$, we get $2$ and $12$. The only two-digit number is $12$, so the answer is $\boxed{\textbf{(B)}\ 12}$.

~TheGoldenRetriever

Video Solution (HOW TO CRITICALLY THINK!!!)

https://youtu.be/EbFw47vZASs

~Education, the Study of Everything


Video Solution

https://youtu.be/PQnA07go4GM

Video Solution

https://youtu.be/zTGuz6EoBWY

See Also

2017 AMC 10B (Problems • Answer Key • Resources)
Preceded by
First Problem 		Followed by
Problem 2
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 10 Problems and Solutions

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