SECTIONC

Question $7[10$ marks$]-$ COMPULSORY

$[$The fraction of the marks attained for this question and video explanation

determine the fraction of the attained marks to be allocated for questions $8$ and $9.]$

$($a$)$ The Greek Mathematician Euclid developed an algorithm named $"$gcd$"$ to

determine the Greatest Common Divisor $($GCD$).$ By developing an algorithm

for Lowest Common Multiple $($LCM$)$ named "Icm" that does not use the GCD$,$

provide an algorithm $"$gcd$\_$primes$\_$lcm" which given two numbers, uses the

algorithms $"$gcd$"$ and "lcm" and determines the Number of Prime Numbers

between the Greatest Common Divisor and the Least Common Multiple

$($LCM$)$ inclusive.

$($b$)$ Using the algorithm$($s$)$ provided for section $($a$)$ of this question, illustrate how

your algorithm would find the Number of Prime Numbers between the

Greatest Common Divisor and the Least Common Multiple $($LCM$)$ exclusive

of $1925$ and $2695[$and$/$or $7096650$ and $9505210-$ Bonus $3$ marks$].$

$($c$)$ Using a Large Language Model $($LLM$)($such as ChatGPT, Gemini, Claude,

etc.$),$ provide a solution for question $($b$)$ and state with reasons which of the

algorithms is more efficient.

$($d$)$ David Campbell and his wife Samantha Campbell entered into marriage on the

day Samantha celebrated her eighteenth birthday. David and Samantha were

born on a Saturday and Sunday respectively three years apart with David being

older. Use modular arithmetic to explain at what age should David and

Samantha have their first and second children if they aim to have the first child

being born on the same day and month as David $($Saturday$)$ and the second

child being born on the same day and month as Samantha $($Sunday$).$

Question marks$]$

Using any programming language taught in your first year at the university, write a

function false fib$\_$filter$\_v3$ that takes three lists of integers $lst1,lst2$ and $lst3,$ and

returns True if any list is not a list of consecutive numbers within the Fibonacci

sequence. False is returned otherwise.

Question $9[4$ marks $+$ Bonus $5$ marks $($to an assignment maximum of $50$ marks$)]$

Let $f(n)$ be defined by

$f(1)=57$

$f(n)=3125f(\frac{n}{5})+71n^5$

if $n>1$ and $n=5^m,$ where m is a positive integer.

$($a$)$ By using the principles of Recurrence Relation, find a general formula

for $f(n)$

$($b$)$ Hence show that

$f(n)=(n^{5}lo{g}_{5}n).$

$[$Bonus $-5]$

$[$The fraction of the marks attained for this question and video explanation

determine the fraction of the attained marks to be allocated for questions $8$ and $9.]$

$($a$)$ The Greek Mathematician Euclid developed an algorithm named $"$gcd$"$ to

determine the Greatest Common Divisor $($GCD$).$ By developing an algorithm

for Lowest Common Multiple $($LCM$)$ named "lcm" that does not use the GCD$,$

provide an algorithm $"$gcd$\_$primes$\_$lcm" which given two numbers, uses the

algorithms $"$gcd$"$ and "lcm" and determines the Number of Prime Numbers

between the Greatest Common Divisor and the Least Common Multiple

$($LCM$)$ inclusive.

$($b$)$ Using the algorithm$($s$)$ provided for section $($a$)$ of this question, illustrate how

your algorithm would find the Number of Prime Numbers between the

Greatest Common Divisor and the Least Common Multiple $($LCM$)$ exclusive

of $1925$ and $2695[$and$/$or $7096650$ and $9505210-$ Bonus $3$ marks$].$

$($c$)$ Using a Large Language Model $($LLM$)($such as ChatGPT, Gemini, Claude,

etc.$),$ provide a solution for question $($b$)$ and state with reasons which of the

algorithms is more efficient.

$($d$)$ David Campbell and his wife Samantha Campbell entered into marriage on the