# 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.