4. determine all sets $M$ of natural numbers such that for every two (not necessarily different) elements $a, b$ from $M$ also

$$
\frac{a+b}{\operatorname{ggT}(a, b)}
$$

is in $M$.

## Solution
We set $f(a, b)=(a+b) / \operator name{ggT}(a, b)$ for the abbreviation. The solution proceeds in several steps.

(i) Let $a \in M$ be any element. Then $f(a, a)=2 \in M$. Note that $M=\{2\}$ is a solution.

(ii) Suppose $M$ contains another even number. Let $2 m$ be the smallest even number in $M$ with $m \geq 2$. Then $f(2 m, 2)=m+1$ in $M$ and because of $2<$ $m+1<2 m$ and the minimality of $m$ this number is odd. So $M$ contains an odd element.
(iii) Suppose $u \in M$ is odd. Because $f(u, 2)=u+2$, $u+2$ is also in $M$, it follows inductively that $M$ contains all odd numbers $\geq u$. In particular, $3 u \in M$ and therefore

$$
4=f(3 u, u) \in M \quad \Longrightarrow \quad 3=f(4,2) \in M
$$

According to the above argument, $M$ therefore contains all odd numbers $\geq 3$.

(iv) Let $2 m$ be any even number. Since 3 and $3(2 m-1)$ are in $M$ according to (iii), $2 m=f(3(2 m-1), 3) \in M$, therefore $M$ also contains all even numbers. Consequently, $M=\mathbb{N}$ or $M=\mathbb{N} \backslash\{1\}$, and it is easy to see that these are both solutions.