Remarks: Solve the following $5$ questions. For each algorithm, always explain how and why it works. ALWAYS analyze the complexity of your algorithms. In all cases, strive to achieve the fastest possible algorithm. A correct algorithm with slow running time may not receive full credit. Do not use material not taught in class. Solve the questions by yourself.

Very important: DO NOT WRITE PROGRAMS.

Question $1$: Suppose we are given an integer n$,$ and an integer x n$.$ The problem is to list all numbers x$*$i such that x$*$i n$,$ where i starts at $1$ and increases until the largest possible value of i$.$

$($a$)$ Given a specific x$,$ what is the number of operations required for the above pro$-$ cedure as a function of x and n $?$

$($b$)$ Suppose that in the above procedure, x is chosen uniformly at random from the numbers $1,2,$dots,n$.$ What is the expected number of operations for the procedure?