4. Task and order of execution / Завдання та порядок виконання
Source of tasks. The nine tasks below are taken exactly from the assignment. Implement the described behaviour; the input/output examples are authoritative. Each task asks you to explain the algorithm or reasoning you used.
Order of execution
The tasks are grouped into three difficulty bands by the effort each takes, not by their number — so Task 9 (a straightforward evaluation) sits in the Easy band while the proof and totient tasks form the Hard band. Work band by band, testing each task against its example; your score corresponds to the highest band you complete correctly.
Easy tasks (60–74 points)
Task 1 — Parity Checker
Objective: Write a program to determine the parity of a given number.
- Input: Integer:
25 - Output: Parity:
Odd
Instructions: Implement a function that accepts an integer and returns its
parity as "Even" or "Odd". Explain the logic and algorithm used in your
implementation.
Task 2 — Prime Number Checker
Objective: Implement a program that checks if a given number is prime.
- Input: Integer:
17 - Output: Prime Status:
Prime
Instructions: Develop a function that takes an integer input and returns
whether it is "Prime" or "Not Prime". Provide an explanation of the method
used to determine the primality.
Task 3 — GCD Calculator
Objective: Create a program to calculate the Greatest Common Divisor (GCD) of two numbers.
- Input: Two integers:
48, 18 - Output: GCD:
6
Instructions: Implement a function that calculates the GCD of two integers. Explain the algorithm and steps involved in the calculation.
Task 9 — Advanced Function Evaluation
Objective: Implement a program to evaluate complex mathematical functions with considerations to their domains and ranges.
- Input: Function:
f(x) = x^2 + 2x + 1; Value:x = 3 - Output: Evaluated Result:
16
Instructions: The program should evaluate the function, consider its domain and range, and provide an explanation of the evaluation process and results.
Medium tasks (75–89 points)
Task 4 — Prime Factorization
Objective: Write a program to find the prime factorization of a given number.
- Input: Integer:
56 - Output: Prime Factors:
2^3 * 7
Instructions: Implement a function to find the prime factors and their
powers. The program should provide the result in the format
"p1^e1 * p2^e2 * ...".
Task 5 — LCM Calculator
Objective: Create a program to compute the Least Common Multiple (LCM) of two integers.
- Input: Two integers:
15, 20 - Output: LCM:
60
Instructions: Develop a function to calculate the LCM, and explain the method used for the calculation and how it relates to the GCD.
Task 6 — Direct Proof Implementation
Objective: Write a program that demonstrates a direct proof method for a given mathematical statement.
- Input: Statement: “The product of two odd integers is odd.”
- Output: Proof steps and validation
Instructions: Implement an algorithm that validates the statement through direct proof. Explain each step and the logical reasoning behind it.
Hard tasks (90–100 points)
Task 7 — Advanced Number Theory Function
Objective: Implement a program that performs a complex operation involving number theory concepts, like Euler’s Totient Function.
- Input: Integer:
12 - Output: Euler’s Totient Value:
4
Instructions: Implement Euler’s Totient Function and provide an explanation of the algorithm and steps in the computation.
Task 8 — Comprehensive Proof by Induction
Objective: Develop a program that can prove mathematical statements using the principle of mathematical induction.
- Input: Statement: “The sum of the first n odd numbers is n^2 for all positive integers n.”
- Output: Proof steps and validation
Instructions: The program should facilitate proof by induction, demonstrating base case verification, induction hypothesis, and induction step.