Problem:
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=444eight .
Solution:
We are looking for a four-digit palindrome in base 8 that is greater than 7778=51110. Any such palindrome must have the form
(ABBA)8=512A+64B+8B+A=513A+72B.
Since A must be 1, we substitute into the expression and evaluate values of B:
B=0:B=1:B=2:B=3:…(No more values under 1000 work)513+72⋅0=513513+72⋅1=585513+72⋅2=657513+72⋅3=729
Since 585 is the only palindrome under 1000 of the form (ABBA)8, the answer is
585.
The problems on this page are the property of the MAA's American Mathematics Competitions