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 a

assuming you're not counting 0....

 

4 is the smallest 5 digit base 2 palindrome, 10 is the 2nd smallest.

 

in base 3 10 is the length 3 palindrome 101

 

 

There are only 5    5 digit palindromes in base 2. They are

10001 base 2 which equals   17 base 10

10101  base 2 which equals   21 base 10

11011  base 2 which equals    27  base 10

11111  base 2 which equals     31  base 10

 

Which of these can be expressed as a palindrome in any base?   Let the base be x

which is of the form   ax^2+bx+a   (base 10)

                 where    a b and x are all  integers 

                 and a and b are both smaller than x

 

x=2 is too small

x=3, a=2       9a+b+a  =   10a+b  = 20+1  works        212 (base3) = 10101 (base2) = 21 (base 10)

 

So the question becomes... can I make 17 with a 3 digit palindrome.

 

101 base 4     That is obvious I should have seen it in the first place

 

\(10001_2=17_{10}=101_4\)

 

So the answer is  10001