The sum of the sum of the sum

Relevant wiki: Application of Divisibility Rules

Let $N$ denote the number of digits of the number $5555^{5555}$ . Then $N =\left \lfloor \log_{10} 5555^{5555} + 1 \right \rfloor = \lfloor 5555 \log{10} 5555 + 1 \rfloor = 20802$ .

So the maximum value of $x$ is, $\max(x) = 1 + \underbrace{ 9 + 9 + 9 + \cdots + 9}_{20801\text{ 9's}} = 1 + 9(20801) = 187210$ .

Similarly, the maximum value of $y$ is $\max(y) = 1 + 7+9+9+9+9 = 44$ .

And, the the maximum value of $z$ is $\max(z) = 3+9 = 12$ . Thus $1\leq z \leq 12$ .

By application of divisibility rules , the repeated sum of digits of $5555^{5555}$ is equal to $5555^{5555} \equiv_9 (5555 \bmod 9)^{5555} \equiv_9 2^{5555} \equiv_9 2^{5555 \pmod{ (\phi(9))} } = 2^5 \equiv_9 5 \; .$

The penultimate step follows from Euler's totient function .

Hence, $z \equiv 5 \pmod 9$ . And because $1\leq z \leq 12$ , then $z = \boxed 5$ .