Question

The first $2007$ positive integers are each written in base $3.$ How many of these base- $3$ representations are palindromes? (A palindrome is a number that reads the same forward and backward.)

$100$

$101$

$102$

$103$

$104$

Solution:

A palindrome is fixed by its first half. Counting base- $3$ palindromes by length gives $2$ of length $1$ or $2,$ $6$ of length $3$ or $4,$ $18$ of length $5$ or $6,$ and $54$ of length $7.$

That totals $2+2+6+6+18+18+54=106$ palindromes with at most $7$ digits. Since $2007=2202100_3,$ the $7$ -digit palindromes larger than it are $2210122,$ $2211122,$ $2212122,$ $2220222,$ $2221222,$ and $2222222,$ which is $6$ of them.

Therefore the count is $106-6=100.$

Thus, the correct answer is A.

2007 AMC 12B Problem 21

Solution:

Problem 21 in Other Years