# 2019 AMC 8 Problems/Problem 13

## Problem 13

A palindrome is a number that has the same value when read from left to right or from right to left. (For example 12321 is a palindrome.) Let $N$ be the least three-digit integer which is not a palindrome but which is the sum of three distinct two-digit palindromes. What is the sum of the digits of $N$? $\textbf{(A) }2\qquad\textbf{(B) }3\qquad\textbf{(C) }4\qquad\textbf{(D) }5\qquad\textbf{(E) }6$

## Solution 1

All the two digit palindromes are multiples of $11$. The least $3$ digit integer that is the sum of $3$ two digit palindromes is a multiple of $11$. The least $3$ digit multiple of $11$ is $110$. The sum of the digits of $110$ is $1 + 1 + 0 =$ $\boxed{\textbf{(A)}\ 2}$.

~heeeeeeheeeee

## Solution 2

We let the two digit palindromes be $AA$, $BB$, and $CC$, which sum to $11(A+B+C)$. Now, we can let $A+B+C=k$. This means we are looking for the smallest $k$ such that $11k>100$ and $11k$ is not a palindrome. Thus, we test $10$ for $k$, which works so $11k=110$, meaning that the sum requested is $1+1+0=\boxed{\textbf{(A)}\ 2}$. ~smartninja2000

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