AMC 10 2021-fall-b Problem 19

Let $N$ be the positive integer $7777\ldots777$, a $313$-digit number where each digit is a $7$. Let $f(r)$ be the leading digit of the $r{ }$th root of $N$. What is$$f(2) + f(3) + f(4) + f(5)+ f(6)?$$$(\textbf{A})\: 8 (\textbf{B}) \: 9 (\textbf{C}) \: 11 (\textbf{D}) \: 22 (\textbf{E}) \: 29$