Greatest Integer and Fractional Part Function

Find the number of positive integers $n$ such that the highest power of $7$ dividing $n!$ is $8.$

For any real number $x,$ let $\left[x\right]$ denote the integer part of $x\mathit{;}\mathit{ }\left\{x\right\}$ be the fractional part of $x\mathit{.}$ $\left(\left\{x\right\}=x-\left[x\right]\right)$. Let $A$ denote the set of all real numbers $x$ satisfying $\left\{x\right\}=\frac{x+\left[x\right]+\left[x+\left({\displaystyle \frac{1}{2}}\right)\right]}{20}$. If $S$ is the sum of all numbers in $A,$ find $\left[S\right]$.

Consider the sequence of numbers $\left[n+\sqrt{2n}+\frac{1}{2}\right]$ for $n\ge 1,$ where $\left[x\right]$ denotes the greatest integer not exceeding $x$. If the missing integers in the sequence are $n_1<n_2<n_3<\dots ..,$ then find $n_{12}$.