Python Wrapup This is a revision of problem set 7 which previously was due on Thursday November 15. As you will see, this problem set absorbs the material in what had been posted as the extra credit problem set 8. NB that the new due date is Wednesday November 21. Like some of the earlier problem sets, this problem set allows you to choose which problems you would like to work on. This is a 15 point problem set but the problems listed here can total up to 36 points. As before, you can do more of the problems for extra credit. I encourage you to carefully review the python code that has been posted on the resources page and as solutions to problem sets 5 and 6. (3 Points) Write a python function factors that accepts an integer n and which returns a list of all integer factors of n. For example, the expression factors(18) could evaluate to the list [1, 18, 2, 9, 3, 6]. (1 Point) Write a python function isPrime that accepts an integer n and which returns True if n is a prime number and False otherwise. For example, isPrime(13) should evaluate to True because the only factors of 13 are 1 and 13. (2 Points) Integers m and n are relatively prime if the only factor that they share is 1. Write a function areRelativelyPrime which accepts two integers and returns True if they are relatively prime. Otherwise it should return False. (3 Points) Write a function gcd which accepts two integers m and n and which computes the greatest common divisor of m and n. For example, gcd(15,20) should evaluate to 5 and gcd(15,17) should evaluate to 1. (1 Point) Write a python function nth which accepts an integer n and a list l and which returns the nth element of l. For example, the expression nth(3,[10,20,30,40]) should evaluate to 30. NB that for the purposes of this problem, the first element of the list is considered at position 1. If n is greater than the length of l your nth function should return 0. (2 Points) Write a python function match which accepts two lists a and b and two integers ai and bi. match should return True if the list a matches list b starting at positions ai and bi (resp.). For example, the expression match([2,3,4,3], 1, [3,4,3,5], 0) should evaluate to True. (3 Points) Write a python function isSublist which accepts two lists a and b and returns True if a is a sublist of b. For example, isSublist([2,3,4], [2,3,2,3,4,5]) should return True while isSublist([2,3,4], [2,3,2,3,5]) should return False. (3 Points) Write a python function rotateRight which accepts an integer n and a list lst. The function should return the list obtained by rotating the lst to the right n times. For example, rotateRight(3,['A', 'B', 'C', 'D', 'E']) should return the list ['C', 'D', 'E', 'B', 'A']. (3 Points) Write a python function merge which accepts two lists of integers lst1 and lst2 in non-decreasing order and which returns a single non-decreasing list of integers combining the two. For example, the expression merge([2,3,3,8], [1,8,9]) should evaluate to the list [1,2,3,3,8,8,9]. (3 Points) Write a python function uniquify that accepts a list and returns the list obtained by removing all copies. For example, uniquify([1,2,3,2,3,4]) should return the list [1,2,3,4]. (1 Point) Write a python function bar that accepts two integers m and n and which uses the turtle graphics module to draw a verticle bar depicting the ratio m/n. For example, the call of the function bar(1,10) should draw a bar extending 1/10 of the distance from the bottom edge of the window to the top. The call bar(10,20) should cause the bar to extend halfway up and, e.g., bar(33,33) should cause the bar to extend all the way to the top. The width and color of the bar are up to you. (3 Points) Recall that the expected frequency of the totals 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 or 12 obtained by rolling 2 dice are easily computed by counting the various combinations that result in each value. There is only one way to roll a 2, there are 2 ways to roll a 3 and so forth. Write a function diceExpectations which uses the turtle graphics module to draw a histogram depicting the expected frequencies of rolling two dice. In particular, the diceExpectations function should draw 11 verticle bars depicting the expected frequencies. The width of each bar should be 1/11 the window width and the height of each bar should be determined by the expected frequency. The height of the bar for 7 (i.e., the middle bar) should be 75% of the height of the window and the height of the other bars should be calibrated relative to that. For example, your histogram might look as follows: (8 Points) Recall that the random module contains a random function which returns a number greater than or equal to 0 and less than 1. The expression: int(random.random() * m) will then evaluate to an integer in the range from 0 up to m - 1. Thus, the expression int(random.random() * 6) + 1 can be used to simulate the roll of a die. Write a python function manyRolls which accepts an integer n and which simulates n rolls of two dice. The function should compute the total of the two dice and count up the number of times each outcome occurs. (I.e., how many times did 2 appear, how many times did 3 appear and so forth.) After completing the n rolls, the manyRolls function should draw two histograms depicting the outcome of the rolls. In particular, the manyRolls function should use the diceExpectations function to draw the histogram for the expected outcome and it should then overlay a new histogram over the old one showing the outcome of the simulation. For example, the histogram on the left reflects 10 rolls while the one on the right reflects 100,000 rolls. Your histogram doesn't have to look exactly like these, but it should allow one to see how well the n simulations matches the expected outcome.