Lab 05

---

Directions

For this lab, you will create a single .R file called lab05.R. The following exercises will ask you to write code. Place all requested code in this .R file separated by comments which indicate which code corresponds to which exercise.

• Note: Do not write your name or netID anywhere in the file or in the filename. To the best of our ability, we will try to grade labs anonymously.

Submit your lab to the corresponding assignment on Canvas. You have unlimited attempts before the deadline. Your final submission before the deadline will be graded.

---

Grading

This lab will be graded based on a mix of correctness and completion. Each exercise is worth 3 points. For each exercise that you demonstrate a good-faith effort to complete, you will receive at least one point. Because there are only two exercises, by turning in the lab you will start with four points.

---

Exercise 1 (Perfect, Abundant, and Deficient Numbers)

For this exercise, you will write four functions:

• find_divisors(x)
• is_perfect(x)
• is_abundant(x)
• is_deficient(x)

Each function takes as input a numeric vector of length 1. You do not need to check that this is the case, but the solution will contain code that checks for these conditions. We’ll be learning how to do this soon.

• The find_divisors function should return a numeric vector containing all of the (proper) divisors of x. These include 1 and exclude x itself.
  • If x is 1, return NULL.
• The is_perfect function should return a logical vector of length 1 that is TRUE if x is perfect, FALSE otherwise.
  • A number \(x\) is perfect if the sum of its proper divisors is equal to \(x\).
• The is_abundant function should return a logical vector of length 1 that is TRUE if x is abundant, FALSE otherwise.
  • A number \(x\) is abundant if the sum of its proper divisors is greater than \(x\).
• The is_deficient function should return a logical vector of length 1 that is TRUE if x is deficient, FALSE otherwise.
  • A number \(x\) is deficient if the sum of its proper divisors is less than \(x\).

Test each of your functions with various inputs. Include these in your .R file after defining the functions. Also include the following code which will quickly test your functions for all numbers between 1 and 25.

• Note: Using a for loop is acceptable, however, not necessary. Also, consider investigating the which() function.

data.frame(
  x = 1:25,
  perfect = sapply(1:25, is_perfect),
  abundant = sapply(1:25, is_abundant),
  deficient = sapply(1:25, is_deficient)
)

---

Exercise 2 (Luhn Algorithm)

For this exercise, you will write a function called is_valid which will check if a number (for example a credit card number) is “valid” according to the Luhn algorithm.

The algorithm starts with a positive integer with two or more digits, then does the following:

• Remove and store the last digit. This is called the check digit.
• With the remaining digits, starting from the rightmost digit moving left, double the value of every second digit (including the rightmost digit).
  • Replace the resulting value at each position with the sum of the digits of this position’s value.
  • Sum up the resulting values from all positions, call this sum s.
• If 10 - (s %% 10) is equal to the check digit, the number is valid.

Your function should output TRUE if the input number is valid, and FALSE otherwise. The function has been started for you below. We also provide a helper function called split_digits that will likely be extremely useful. Include both of these functions in your .R file.

split_digits = function(x) {
  
  if (!is.numeric(x) | length(x) != 1 | trunc(x) != x) {
    stop("x stop be a numeric vector (that represents an \"integer\") of length 1")
  }

  return(as.numeric(strsplit(as.character(x), split = "")[[1]]))
  
}

is_valid = function(num) {
  
  if (length(split_digits(x = num)) < 2) {
    stop("input numbermust have at least two digits")
  }
  
  # delete this comment and place your code here
  
}

Also include the following lines in your .R file. Run them to check your function.

is_valid(num = 79927398713)      # should return TRUE
is_valid(num = 4539319503436467) # should return TRUE
is_valid(num = 8273123273520569) # should return FALSE