Please type or neatly write your solutions to these problems. Show all work for full credit!

$(3$ points$)$ Could each of the following statements be proven by exhaustion? Write yes or no$,$ followed by a short explanation for each. You do not have to actually prove or disprove the statement; you're just answering whether or not a proof by exhaustion is theoretically possible.

$($a$)$ Every sloth on Earth $($at a specific instant in time$)$ is adorable.

$($b$)$ There are no positive integers $a,b,$ and $c$ such that $a^5+b^5=c^5.$

$($c$)$ There are no positive integers $n$ that satisfy the equation $n^3+5n^2=1743.($Hint: Can you determine a maximum possible value for $n$ somehow?$)$