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Difference between revisions of "2010 AMC 12A Problems/Problem 6"

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===Solution 3===
 
===Solution 3===
 
Since we know <math>x+32</math> to be <math>1 a a 1</math> and the only palindrome that works is <math>0 = a</math>, that means <math>x+32 = 1001</math>, and so <math>x = 1001 - 32 = 969</math>. So, <math>9</math> + <math>6</math> + <math>9</math> = <math>\boxed{\textbf{(E)}\ 24}</math>.
 
Since we know <math>x+32</math> to be <math>1 a a 1</math> and the only palindrome that works is <math>0 = a</math>, that means <math>x+32 = 1001</math>, and so <math>x = 1001 - 32 = 969</math>. So, <math>9</math> + <math>6</math> + <math>9</math> = <math>\boxed{\textbf{(E)}\ 24}</math>.
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~songmath20
  
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== Video Solution ==
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== Video Solution by OmegaLearn ==
 
https://youtu.be/ZhAZ1oPe5Ds?t=1444
 
https://youtu.be/ZhAZ1oPe5Ds?t=1444
  
 
~ pi_is_3.14
 
~ pi_is_3.14
  
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==Video Solution by the Beauty of Math==
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== Video Solution ==
−
https://www.youtube.com/watch?v=P7rGLXp_6es
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https://youtu.be/IQj27LEQF4Y
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~Education, the Study of Everything
  
 
== See also ==
 
== See also ==

Latest revision as of 03:51, 21 January 2023

The following problem is from both the 2010 AMC 12A #6 and 2010 AMC 10A #9, so both problems redirect to this page.

Problem

A $\text{palindrome}$, such as $83438$, is a number that remains the same when its digits are reversed. The numbers $x$ and $x+32$ are three-digit and four-digit palindromes, respectively. What is the sum of the digits of $x$?

$\textbf{(A)}\ 20 \qquad \textbf{(B)}\ 21 \qquad \textbf{(C)}\ 22 \qquad \textbf{(D)}\ 23 \qquad \textbf{(E)}\ 24$

Solution

Solution 1

$x$ is at most $999$, so $x+32$ is at most $1031$. The minimum value of $x+32$ is $1000$. However, the only palindrome between $1000$ and $1032$ is $1001$, which means that $x+32$ must be $1001$.

It follows that $x$ is $969$, so the sum of the digits is $\boxed{\textbf{(E)}\ 24}$.

Solution 2

For $x+32$ to be a four-digit number, $x$ is in between $968$ and $999$. The palindromes in this range are $969$, $979$, $989$, and $999$, so the sum of the digits of $x$ can be $24$, $25$, $26$, or $27$. Only $\boxed{\textbf{(E)}\ 24}$ is an option, and upon checking, $x+32=1001$ is indeed a palindrome.

Solution 3

Since we know $x+32$ to be $1 a a 1$ and the only palindrome that works is $0 = a$, that means $x+32 = 1001$, and so $x = 1001 - 32 = 969$. So, $9$ + $6$ + $9$ = $\boxed{\textbf{(E)}\ 24}$. ~songmath20

Video Solution by OmegaLearn

https://youtu.be/ZhAZ1oPe5Ds?t=1444

~ pi_is_3.14

Video Solution

https://youtu.be/IQj27LEQF4Y

~Education, the Study of Everything

See also

2010 AMC 12A (Problems • Answer Key • Resources)
Preceded by
Problem 5 		Followed by
Problem 7
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
2010 AMC 10A (Problems • Answer Key • Resources)
Preceded by
Problem 8 		Followed by
Problem 10
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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