We use cookies to personalise content and advertisements and to analyse access to our website. Furthermore, our partners for online advertising receive pseudonymised information about your use of our website.
Please click on "Accept cookies" if you agree to the setting of cookies. Cookies that do not require consent remain unaffected by this, see
cookie policy and privacy policy.
DECLINE COOKIES

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.

ant101 Sep 4, 2018

#1**+1 **

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

Rom Sep 5, 2018

#4**+1 **

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

Melody Sep 9, 2018