While iterations P30196

You have to program several functions. Do not use the math module.

1. Write an integer function @int_root(n)@ that given a natural number $n$ returns $\lfloor\sqrt{n}\rfloor$.

2. Write a function @int_log(a, b)@ that given natural numbers $a$ greater than one and $b$ greater than zero returns natural $k$ such that $a^k \le b < a^{k + 1}$.

3. Write a function @gcd_lcm(a, b)@ that given natural numbers $a$ and $b$ such that $a\not = 0$ or $b\not =0$ returns the greatest common divisor and the least common multiple. Your code has to implement the Euclid’s algorithm.

4. Write a boolean function @is_prime(n)@ that given a natural number $n$ returns True if and only if $n$ is prime.

5. In order to play table games at the casino you need some tokens. Red tokens cost $7$ euros and yellow tokens cost $4$. Write a function @buy_tokens(n)@ that given a number $n$ of euros such that $n \ge 20$, it returns the equivalence in tokens. When several equivalences are possible the function returns the one minimizing the total number of tokens.

6. Write a string function @max_overlap(s, t)@ that given two strings $s$ and $t$ returns the longest string that is a common prefix of $s$ and $t$.

Scoring

The first function counts 15 points. Other ones count 17 point each one.

Sample session