\section*{Problem 5 - 011025}

Given is the number \(9^{\left(9^{9}\right)}\).

a) How many digits does this number have approximately? (Round to four significant figures.)

b) How long would the strip need to be if one wanted to print this number, assuming the width of each digit is \(2 \mathrm{~mm}\)?

c) With which digit does the sought number end?
}

a) The number of digits is equal to the smallest whole number greater than

\[
\log _{10} 9^{\left(9^{9}\right)}=9^{9} \cdot \log _{10} 9
\]

Given \(9^{9}=387420489\) and \(\log _{10} 9 \approx 0.954243\), \(9^{\left(9^{9}\right)}\) thus has approximately 369,700,000 digits.

b) Such a strip would need to be approximately \(369,700,000 \cdot 2 \text{ mm} = 739,400,000 \text{ mm} = 739.4 \text{ km}\) long.

c) Since \(9^{9}\) is an odd number, the number ends in the digit 9 because

\[
9^{\left(9^{9}\right)} \equiv(-1)^{\left(9^{9}\right)} \equiv-1 \equiv 9 \bmod 10
\]