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8. (i)

a) Describe how an imperative programming language executes. State one example of imperative programming language.
An imperative programming language executes by providing a sequence of explicit instructions that tell the computer how to perform a task. These instructions modify the program's state (the values stored in memory) step-by-step. The execution follows a defined control flow, often involving loops, conditional statements, and function calls, directly manipulating data.
Example: Python (when used in a procedural style). Other examples include C, C++, Java.

b) What is variable declaration and what type of programming error arises when one fails to declare a variable?
Variable declaration is the process of defining a variable's name and its data type (e.g., integer, string, boolean) before it is used in a program. This allocates memory for the variable and informs the compiler/interpreter about the kind of values it will store.
When one fails to declare a variable, a syntax error or compilation error (specifically, an "undeclared variable" or "name error") typically arises, as the program cannot recognize or allocate memory for an unknown identifier.

c) State two advantages and two disadvantages of programming using imperative high-level languages as opposed to low-level languages.
Advantages of High-Level Languages:

  1. Readability and Maintainability: High-level languages use syntax closer to human language, making code easier to read, write, and maintain.
  2. Portability: Programs written in high-level languages are generally more portable, meaning they can run on different hardware platforms with minimal or no changes.

Disadvantages of High-Level Languages:

  1. Less Control over Hardware: High-level languages abstract away hardware details, offering less direct control over memory management and processor operations.
  2. Slower Execution Speed: Programs often execute slower than low-level languages because they require compilation or interpretation, which adds an overhead.

8. (ii)

a) Determine the values of enigma(4, 6) and mystery(6, 4).

For enigma(4, 6):

  1. Call enigma(4, 6).

    Since n=6≠0n=6 \neq 0, it calls enigma(6, 4 mod 6).

    4 mod 6=44 \bmod 6 = 4

    So, it calls enigma(6, 4).

  2. Call enigma(6, 4).

    Since n=4≠0n=4 \neq 0, it calls enigma(4, 6 mod 4).

    6 mod 4=26 \bmod 4 = 2

    So, it calls enigma(4, 2).

  3. Call enigma(4, 2).

    Since n=2≠0n=2 \neq 0, it calls enigma(2, 4 mod 2).

    4 mod 2=04 \bmod 2 = 0

    So, it calls enigma(2, 0).

  4. Call enigma(2, 0).

    Since n=0n=0, it returns mm.

    m=2m = 2

    Therefore, enigma(4, 6) returns 2.

    2\boxed{2}

    For mystery(6, 4):

  5. Call mystery(6, 4).

    Since x=6≠0x=6 \neq 0 and y=4≠0y=4 \neq 0, it executes the else block.
    It returns (x * y) / enigma(x, y), which is (6 * 4) / enigma(6, 4).

  6. First, evaluate enigma(6, 4)

    • Call enigma(6, 4). Since n=4≠0n=4 \neq 0, it calls enigma(4, 6 mod 4). 6 mod 4=26 \bmod 4 = 2 So, it calls enigma(4, 2).
    • Call enigma(4, 2). Since n=2≠0n=2 \neq 0, it calls enigma(2, 4 mod 2). 4 mod 2=04 \bmod 2 = 0 So, it calls enigma(2, 0).
    • Call enigma(2, 0). Since n=0n=0, it returns mm. m=2m = 2

    So, enigma(6, 4) returns 2.

  7. Substitute the result back into the mystery calculation.

    mystery(6,4)=(6×4)/2mystery(6, 4) = (6 \times 4) / 2 mystery(6,4)=24/2mystery(6, 4) = 24 / 2 mystery(6,4)=12mystery(6, 4) = 12

    Therefore, mystery(6, 4) returns 12.

    12\boxed{12}

    b) What does each of the functions Enigma and Mystery computes?

    • The function enigma(m, n) computes the Greatest Common Divisor (GCD) of two positive integers m and n, using the Euclidean algorithm.
    • The function mystery(x, y) computes the Least Common Multiple (LCM) of two positive integers x and y. It does this by using the formula: LCM(x,y)=(x×y)/GCD(x,y)\text{LCM}(x, y) = (x \times y) / \text{GCD}(x, y).

    c) Which of the above functions is recursive? State the reason.
    The function enigma is recursive.
    Reason: A function is recursive if it calls itself within its own definition. In enigma, the else branch contains the line return enigma(n, m mod n), which is a call to enigma itself.

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