Problem $1.$ A prime number $($or a prime$)$ is a natural number that has exactly two distinct natural number divisors: $1$ and itself. The purpose of this problem is to write a function $($say its name is check.prime$)$ to check whether or not a given natural number is a prime. Unless you want to use some other more advanced method, you can write your function based on the so$-$called "trial division" method. The idea is as follows. For a positive integer n$,$ one only needs to test whether $n$ is a multiple of each integer $m$ between $2$ and $\sqrt[\phantom{2}]{n}.$ If $n$ is a multiple of any of these integers then it is a composite number and not a prime; if it is not a multiple of any of these integers then it is a prime. Of course, you can use other methods. For example, you can check how many factors $n$ has. If $n$ has exactly $2$ factors $($must be positive integers less than or equal to $n),$ then $n$ is a prime.

The input of check.prime should be a positive integer and the output should be a list containing a logic variable called result and a character variable called report, which tells if the input is a prime. Specific requirements are implied from the following examples.

a$.$ If one feeds anything but a positive integer, assign NA to result and report will be "Error! The input must be a positive integer".

b$.$ We know $37$ is a prime number. Implementing check.prime$(37)$ will yield a T or TRUE for result and "The input $37$ IS a prime number." for report.

c$.$ Implementing check.prime$(51)$ will yield a F or FALSE for result and "The input $51$ IS NOT a prime number." for report.

Complete the following tasks and present your output.

$($i$).$ Implement your R function with $5,21,47\mathrm{.}5,63,87,$ and $111$ one by one and show your results.

$($ii$).$ Implement $x=flr(100**runif(1))$; check.prime$(x)$ for three times and display your ontput.