NYU-CS Homework #7 Solution

Submission instructions:
1. For this assignment, you should turn in 5 files:
• Two ‘.cpp’ files, one for each question 1-2.
Name your files ‘YourNetID_hw7_q1.cpp’, and ‘YourNetID_hw7_q2.cpp’.
• One ‘.pdf‘ file with your answers for questions 3-7. Each question should start on a new page!
Name your file ‘YourNetID_hw7_q3to7.pdf’

2. You must type all your solutions. We will take off points for submissions that are handwritten.

3. You should submit your homework in the Gradescope system.
Note that when submitting the pdf file, you would be asked to assign the pages from your file to their corresponding questions.

4. You can work and submit in groups of up to 4 people. If submitting as a group, make sure to associate all group members to the submission on gradescope.

5. Pay special attention to the style of your code. Indent your code correctly, choose meaningful names for your variables, define constants where needed, choose the most appropriate control flow statements, break down your solutions by defining functions, etc.

6. For the math questions, you are expected to justify all your answers, not just to give the final answer (unless explicitly asked to).
As a rule of thumb, for questions taken from zyBooks, the format of your answers, should be like the format demonstrated in the sample solutions we exposed.


Question 1:
a. Implement a function:
int printMonthCalender(int numOfDays, int startingDay)
This function is given two parameters:
• numOfDays - The number of days in the month

The function should:
• Print a formatted monthly calendar of that month
• Return a number 1-7 that represents the day in the week of the last day in that month.
Formatting Notes:
• The output should include a header line with the days’ names.
• Columns should be spaced by a Tab.

Example: when calling printMonthCalender(31, 4)it should return 6, and should print:
Mon Tue Wed Thr Fri Sat Sun
1 2 3 4
5 6 7 8 9 10 11
12 13 14 15 16 17 18
19 20 21 22 23 24 25
26 27 28 29 30 31

b. A method for determining if a year is a leap year in the Gregorian calendar system is to check if it is divisible by 4 but not by 100, unless it is also divisible by 400.
For example, 1896, 1904, and 2000 were leap years but 1900 was not.
Write a function that takes in a year as input and return true if the year is a leap year, return false otherwise.
Note: background on leap year https://en.wikipedia.org/wiki/Leap_year

c. Implement a function:
void printYearCalender(int year, int startingDay)
This function is given two parameters:

The function should use the functions from sections (a) and (b) in order to print a formatted yearly calendar of that year.



d. Write program that interacts with the user and your function in (c).



Question 2:
Consider the following definitions:
a. A proper divisors of a positive integer (≥ 2) is any of its divisors excluding the number itself. For example, the proper divisors of 10 are: 1, 2 and 5.
b. A perfect number is a positive integer (≥ 2) that is equal to the sum of its proper divisors. For example, 6 and 28 are perfect numbers, since:
6 = 1 + 2 + 3
28 = 1 + 2 + 4 + 7 + 14
Background of perfect numbers: https://en.wikipedia.org/wiki/Perfect_number
c. Amicable numbers are two different positive integer (≥ 2), so related that the sum of the proper divisors of each is equal to the other number. For example, 220 and 284 are amicable numbers, since:
284 = 1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110

220 = 1 + 2 + 4 + 71 + 142


Background of amicable numbers: https://en.wikipedia.org/wiki/Amicable_numbers


a. Write a function:
void analyzeDividors(int num, int& outCountDivs, int& outSumDivs)
The function takes as an input a positive integer num (≥ 2), and updates two output parameters with the number of num's proper divisors and their sum.
For example, if this function is called with num=12, since 1, 2, 3, 4 and 6 are 12s proper divisors, the function would update the output parameters with the numbers 5 and 16. Note: Pay attention to the running time of your function. An efficient implementation would run in Θ"√𝑛𝑢𝑚’.

b. Use the function you wrote in section (a), to implement the function: bool isPerfect(int num)
This functions is given positive integer num (≥ 2), and determines if it is perfect number or not.

c. Use the functions you implemented in sections (a) and (b), to write a program that reads from the user a positive integer M (≥ 2), and prints:
• All the perfect numbers between 2 and M.
• All pairs of amicable numbers that are between 2 and M (both numbers must be in the range).
Note: Pay attention to the running time of your implementation. An efficient algorithm for this part would call analyzeDividors Θ(𝑀) times all together.





Question 3:
a. Solve Exercise 8.2.2, section b from the Discrete Math zyBook.
b. Solve Exercise 8.3.5, sections a-e from the Discrete Math zyBook

Question 4:
Solve the following questions from the Discrete Math zyBook: a) Exercise 5.1.2, sections b, c
b) Exercise 5.3.2, section a
c) Exercise 5.3.3, sections b, c
d) Exercise 5.2.3, sections a, b

Question 5:
Solve the following questions from the Discrete Math zyBook: a) Exercise 5.4.2, sections a, b
b) Exercise 5.5.3, sections a-g
c) Exercise 5.5.5, section a
d) Exercise 5.5.8, sections c-f
e) Exercise 5.6.6, sections a, b

Question 6:
Solve the following questions from the Discrete Math zyBook: a) Exercise 5.7.2, sections a, b
b) Exercise 5.8.4, sections a, b

Question 7:
How many one-to-one functions are there from a set with five elements to sets with the following number of elements?
a) 4
b) 5
c) 6
d) 7


Appendix A.

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