Tutorial questions

1. (Easy)

2. (1)
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   Compute the sum of the first $100$ numbers using a for loop. Can you also do it using a while loop?

3. (2)
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   (*) Write a GAP function that computes the $n$th power of a given number. Specifically, the function should take a number $x$ and an integer $n$, and return $x^{n}$ without using the operator ^. Can you write a non-recursive solution without using ^?

4. (3)
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   Given a list of integers, return the sublist consisting of those that are powers of two or three.

5. (4)
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   Define a function such that, given an integer $n$, it returns $3n$ if $n\equiv 0\pmod{3}$, $(n-1)^{3}$ if $n\equiv 1\pmod{3}$, and $0$ otherwise. Compute the value of such a function on the interval $[-100,100]$. How many times do you get a $0$ as an answer?

6. (Medium)

7. (5)
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   (*) Write a GAP function that takes a matrix with integer coefficients as input and returns a new matrix where:

   • •
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     The odd coefficients are doubled.

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     The even coefficients remain unchanged.

8. (6)
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   Return the first ten pairs $(p,q)$ of prime numbers satisfying $q=p+2$.

9. (7)
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   (*) The Collatz conjecture.

   1. (a)
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      Define the “Collatz" function $f_{C}:\mathbb{N}\to\mathbb{N}$.

   2. (b)
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      Define a function such that given a positive integer $n$, it returns $k$ such that $f_{C}^{k}(n)=1$. Here, $f_{C}^{k}$ denotes the composition of $f_{C}$ with itself $k$ times. It is conjectured that such a $k$ always exists for any positive integer.

10. (Hard)

11. (8)
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    (*) Find (if possible) the smallest solution of

    $$\begin{cases}x\equiv 3\bmod 10,\\ x\equiv 8\bmod 15,\\ x\equiv 5\bmod 84.\end{cases}$$

    Hint: Use the Chinese reminder theorem.

    Hint 2: Use the documentation.

12. (9)
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    The Look and Say sequence. Look at this sequence of numbers.

    $\begin{array}[]{l}1\\ 11\\ 21\\ 1211\\ 111221\\ 312211\\ 13112221\end{array}$

    Can you guess the function that computes the first 100 terms?

    Compare your answer with https://oeis.org/A005150.

More exercises

1. (9)
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   Given a list of integers, return the sublist consisting of those that are powers of two or three.

2. (10)
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   Define the Euler Totient function. Remember that Euler Totient function (often denoted as $\phi(n)$) counts the number of positive integers up to $n$ that are coprime to $n$.

3. (11)
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   Define a function that inputs a positive integer and returns its double factorial.

4. (12)
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   Define a function such that, given a string, it returns fail if the string is empty, and replaces the last character of the string by the symbol $\&$ otherwise.

5. (13)
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   Define a function such that, given a list, it returns a new list obtained by reversing the second half of the input list.

6. (14)
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   Define a function such that, given two non-empty lists, it returns a new list obtained by interlacing the values of the input lists.

7. (15)
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   Given a set of elements $S$ in a given group $G$, return a smaller subset of $S$ consisting of $G$-conjugate representatives (lying in $S$).

   Hint: Use IsConjugate.