### Instruction: 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.### Response: To start, I know that the smallest 5-digit palindrome in base 2 is 10001, which is 17 in base 10.
I wonder if 17 is a palindrome in any other base.
I can try some small bases, like base 3, 4, and 5.
In base 3, 17 is 122, which is not a palindrome.
In base 4, 17 is 101, which is a palindrome.
So I have found a possible answer, but I want to check if there is a smaller one.
The next smallest 5-digit palindrome in base 2 is 100101, which is 37 in base 10.
I can repeat the process and try some small bases.
In base 3, 37 is 1101, which is not a palindrome.
In base 4, 37 is 211, which is not a palindrome.
In base 5, 37 is 122, which is not a palindrome.
In base 6, 37 is 101, which is a palindrome.
So I have found another answer, but it is larger than the previous one.
Therefore, the smallest 5-digit palindrome in base 2 that can be expressed as a 3-digit palindrome in a different base is 10001, which is 101 in base 4.