# Difference between revisions of "2010 AMC 10A Problems/Problem 10"

## Problem 9

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

## Solution $\boxed{(E)}$ $2017$

There are $365$ days in a non-leap year. There are $7$ days in a week. Since $365 = 52 \cdot 7 + 1$ (or $365$ is congruent to $1 \mod{ 7}$), the same date (after February) moves "forward" one day in the subsequent year, if that year is not a leap year.

For example: $5/27/08$ Tue $5/27/09$ Wed

However, a leap year has $366$ days, and $366 = 52 \cdot 7 + 2$ . So the same date (after February) moves "forward" two days in the subsequent year, if that year is a leap year.

For example: $5/27/11$ Fri $5/27/12$ Sun

You can keep count forward to find that the first time this date falls on a Saturday is in $2017$: $5/27/13$ Mon $5/27/14$ Tue $5/27/15$ Wed $5/27/16$ Fri $5/27/17$ Sat