Question

Recall that a palindrome is a number that reads the same forward and backward. Find the greatest integer less than $1000$ that is a palindrome both when written in base ten and when written in base eight, such as $292 = 444_{\text{eight}}.$

Answer

Solution:

A four-digit base-eight number lies between $512$ and $4095,$ so a base-eight palindrome less than $1000$ with four digits must have leading (and trailing) digit $1:$ it has the form $\overline{1bb1}_{\text{eight}} = 512 + 64b + 8b + 1 = 513 + 72b.$ Keeping this below $1000$ requires $b \le 6,$ giving the candidates $513, 585, 657, 729, 801, 873, 945.$

Checking from the top, the only one of these that is also a palindrome in base ten is $585 = 1111_{\text{eight}}.$ Every base-eight palindrome with at most three digits is at most $777_{\text{eight}} = 511 \lt 585,$ so the answer is $585.$

2023 AIME II Problem 2

Solution:

Problem 2 in Other Years