# 2010 AMC 12B Problems/Problem 11

The following problem is from both the 2010 AMC 12B #11 and 2010 AMC 10B #21, so both problems redirect to this page.

## Problem

A palindrome between $1000$ and $10,000$ is chosen at random. What is the probability that it is divisible by $7$?

$\textbf{(A)}\ \dfrac{1}{10} \qquad \textbf{(B)}\ \dfrac{1}{9} \qquad \textbf{(C)}\ \dfrac{1}{7} \qquad \textbf{(D)}\ \dfrac{1}{6} \qquad \textbf{(E)}\ \dfrac{1}{5}$

## Solution 1

View the palindrome as some number with form (decimal representation): $a_3 \cdot 10^3 + a_2 \cdot 10^2 + a_1 \cdot 10 + a_0$. But because the number is a palindrome, $a_3 = a_0, a_2 = a_1$. Recombining this yields $1001a_3 + 110a_2$. 1001 is divisible by 7, which means that as long as $a_2 = 0$, the palindrome will be divisible by 7. This yields 9 palindromes out of 90 ($9 \cdot 10$) possibilities for palindromes. However, if $a_2 = 7$, then this gives another case in which the palindrome is divisible by 7. This adds another 9 palindromes to the list, bringing our total to $18/90 = \boxed {\frac{1}{5} } = \boxed {E}$

## Solution (Divisibility Rules)

We can notice the palindrome is of the form $\overline{abba}$. Then, by the divisibility rule of $7$, $7$ must divide $$100a+11b-2a = 98a+11b.$$ This nicely simplifies to the fact that $7 \mid 4b,$ so $b$ is clearly $0$ or $7$. This gives us $9 \cdot 2$ total choices for the palindrome divisible by $7$, divided by $9 \cdot 10$ total choices for $\overline{abba}$, giving us an answer of $\boxed{\text{(E)}} \ \dfrac{1}{5}$.

~icecreamrolls8

$7\mid 1001a^3+110b^2$ and $1001 \equiv 0 \pmod 7$. Knowing that $a$ does not factor (pun intended) into the problem, note 110's prime factorization and $7\mid b$. There are only 10 possible digits for $b$, 0 through 9, but $7\mid b$ only holds if $b=0, 7$. This is 2 of the 10 digits, so $\frac{2}{10}=\boxed{\textbf{E)}\frac{1}{5}}$

~BJHHar

~IceMatrix