Algorithm A$1$ takes $104\backslash$times n$2$ seconds to solve a problem instance of size n and Algorithm A$2$ takes $102\backslash$times n$3$ seconds to do the same on a particular machine.

$($a$)$ What is the size of the largest problem instance A$1$ will be able to solve in one year?

$($b$)$ What is the size of the largest problem instance A$1$ will be able to solve in one year on a machine one hundred times as fast?

$($c$)$ What is the size of the largest problem instance A$2$ will be able to solve in one year?

$($d$)$ What is the size of the largest problem instance A$2$ will be able to solve in one year on a machine one hundred times as fast?

$($e$)$ Which algorithm will produce results faster, in case we are trying to solve problem instances of size less than $20?$

Please keep in mind that the largest problem instance is an integer.