2010 AMC 12A Problems/Problem 6

Problem 6

A $\texti{palindrome}$ (Error compiling LaTeX. Unknown error_msg), 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

$x$ 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 $\boxed{\textbf{(E)}\ 24}$ .

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2010 AMC 12A Problems/Problem 6

Problem 6

Solution