Cyclic Digital Sum

For an $n$ -digit cyclic number to exist, $n + 1$ must be a prime and $\frac{1}{n + 1}$ must have a digital period of $n$ digits. If these conditions are met, then the $n$ -digit cyclic number consists of the digits in one digital period of $\frac{1}{n + 1}$ . For example, the given $6$ -digit cyclic number exists because $6 + 1 = 7$ is a prime and $\frac{1}{7}$ has a digital period of $6$ digits, and since $\frac{1}{7} = 0.\overline{142857}$ , the $6$ -digit cyclic number is $142857$ .

Furthermore, Midy's Theorem states that the digits in the first half of the cyclic number and the corresponding digits in the second half of the cyclic number add up to 9. (For example, in the $6$ -digit cyclic number of $142857$ , $1 + 8 = 9$ , $4 + 5 = 9$ , and $2 + 7 = 9$ .) This means that the sum of the digits of any $n$ -digit cyclic number is $\frac{9n}{2}$ .

Therefore, a $700$ -digit cyclic number has a digit sum of $\frac{9 \cdot 700}{2} = \boxed{3150}$ .