Question

A five-digit palindrome is a positive integer with respective digits $abcba,$ where $a$ is not zero. Let $S$ be the sum of all five-digit palindromes. What is the sum of the digits of $S?$

$9$

$18$

$27$

$36$

$45$

Answer: B

Concepts:palindrome place value summation

Difficulty rating: 1660

Solution:

Write $\overline{abcba}=10001a+1010b+100c.$ Summing over all palindromes, each value of $a\in\{1,\dots,9\}$ occurs with $10\cdot10=100$ choices of $b,c,$ and each value of $b$ or $c$ occurs with $9\cdot10=90$ choices of the other two digits.

Using $\sum a=\sum b=\sum c=45,$ $\begin{gathered} \scriptsize S=45\big(10001\cdot100+1010\cdot90+100\cdot90\big)\\ =45\cdot1{,}100{,}000\\ =49{,}500{,}000. \end{gathered}$

The sum of the digits of $S$ is $4+9+5=18.$

Thus, the correct answer is B.

2014 AMC 12A Problem 15

Answer: B

Difficulty rating: 1660

Solution:

Problem 15 in Other Years