### 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: I know that the smallest 5-digit palindrome in base 2 is 10001, which is 17 in base 10.
But is 17 a 3-digit palindrome in any other base?
I can try to convert 17 to different bases by dividing by the base and taking the remainders.
If I try base 2, I get 10001 again, which is not 3 digits.
If I try base 3, I get 122, which is not a palindrome.
If I try base 4, I get 101, which is a palindrome.
So 17 in base 10 is 10001 in base 2 and 101 in base 4.
That means the smallest 5-digit palindrome in base 2 that can be expressed as a 3-digit palindrome in a different base is 10001.