## Task 2 - 130722

Karl is looking for three non-zero natural numbers $a, b, c$, for which the following holds:

$(a, b)=4$ (read: The gcd of the numbers $a$ and $b$ is 4),

$(a, c)=6$,

$(b, c)=14$.

After some trial and error, he claims that there are even multiple ways to specify three such numbers.

Is this claim correct?

Is there a way to choose three such numbers $a, b, c$ such that, compared to all other possibilities, $a$ is the smallest and at the same time $b$ is the smallest and at the same time $c$ is the smallest? If so, then give the numbers $a, b, c$ for this case!
If $a, b, c$ satisfy the three mentioned $\operatorname{ggT}$ conditions, then $a$ is divisible by 4 and by 6, and thus by the $\mathrm{kgV}$ of these numbers, i.e., by 12.

Furthermore, $b$ is then divisible by 4 and by 14, and thus by the $\mathrm{kgV}$ of these numbers, i.e., by 28. Similarly, $c$ is divisible by 6 and 14, and thus by 42.

On the other hand, the choice of (1) $a=12, b=28, c=42$ satisfies all three $\operatorname{ggT}$ conditions. Therefore, the second question of the task is to be answered with Yes and the specification (1).

If one multiplies $a$ in (1) by a number $z>1$ that is coprime to $b$ and $c$ (e.g., with $z=5$), one obtains another set of three numbers (in the example 60, 28, 42) that also satisfy all three $\operatorname{ggT}$ conditions. Therefore, the first question of the task is also to be answered with Yes.