Problem $2.$ In the lemon game, $n$ people, numbered $1$ to $n,$ are sitting in a circle. Starting at person $1,$ a lemon

is passed. After $m$ passes, the person holding the lemon is eliminated, the circle closes ranks, and the game

continues with the person who was sitting after the eliminated person picking up the lemon. The last remaining

person wins. Thus, if $m=0$ and $n=4,$ players are eliminated in order, and player $4$ wins. If $m=1$ and $n=5,$

the order of elimination is $2,4,1,5.$

$($a$)$ Write a program to solve the lemon problem for general values of $m$ and $n.$ Try to make your program as

efficient as possible. Make sure you dispose of cells.

$($b$)$ What is the running time of your program?