#### TheDivideByNineMethod

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Let $m$ be the least positive integer divisible by $17$ whose digits sum is $17$. Find $m$.

The number $2^{29}$ is a nine-digit number whose digits are all distinct. Which digit of $0$ to $9$ does not appear?

Let $f(n)$ denote the sum of the digits of $n$. Find $f(f(f(4444^{4444})))$.

Let $S(n)$ equal the sum of the digits of positive integer $n$. For example, $S(1507) = 13$. For a particular positive integer $n$, $S(n) = 1274$. Which of the following could be the value of $S(n+1)$?

The two-digit integers from $19$ to $92$ are written consecutively to form the large integer $$N=192021\cdots 909192$$

Suppose that the $3^k$ is the highest power of $3$ that is a factor of $N$. What is $k$.

Select nine different digits from $0$ to $9$ to form a two-digit number, a three-digit number and a four-digit number. The sum of these three numbers is $2017$. Which digit is not selected?

Let $n$ be a positive integer and function $S_1(n)$ return the square of the sum of $n$'s digits. Additionally, let $S_{k+1}(n)=S_1\left(S_k(n)\right)$, where $k$ is a positive integer. Find the value of $S_{1991}(2^{1990})$.

Let $N=4568^{7777}$, $a$ be the sum of digits in $N$, $b$ be the sum of digits in $a$, and $c$ be the sum of digits in $b$. Find $c$.