\section*{Exercise 3 - 261023}

We typically represent numbers in the decimal positional system (using the base 10 and the digits \(0,1, \ldots, 9\)).

Numbers can also be represented in the binary positional system (or binary system) using the base 2 and the digits 0 and 1. To distinguish this binary representation of a number, it is marked with square brackets and a small 2 appended.

a) Provide the binary representation for the number 47!

Determine the representation in the decimal positional system for the number whose binary representation is \([110001]_{2}\)!

b) A natural number is called a decimal palindrome if its decimal representation reads the same sequence of digits from right to left as from left to right.

Determine at least two natural numbers greater than 9 that have the property of being both decimal and binary palindromes!
}

a) It is

\[
47=32+8+4+2+1=2^{5}+2^{3}+2^{2}+2^{1}+2^{0}=[101111]_{2}
\]

For \([110001]_{2}\) we get

\[
[110001]_{2}=2^{0}+2^{4}+2^{5}=48
\]

b) Systematic trial of two-digit palindromic numbers 11, 22, ... yields numbers that are palindromic in both decimal and binary:

\[
33=[100001]_{2} \quad ; \quad 99=[1100011]_{2}
\]