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jmc

number theory senior

Problem

A palindrome is a number that reads the same forward and backward. What is the smallest 5-digit palindrome in base 2 that can be expressed as a 3-digit palindrome in a different base? Give your response in base 2.
Solution
The smallest possible 5-digit palindrome in base 2 is , which is . Now we try converting 17 to other bases. In base 3, we get , and in base 4, we get , which is a palindrome. So works.
Final answer
10001_2