On the table are written three natural numbers $x, y$, and $z$. Determine which, if you know that simultaneously:

- $x$ is the largest of them,
- the least common multiple of numbers $x$ and $y$ is 200,
- the least common multiple of numbers $y$ and $z$ is 300,
- the least common multiple of numbers $x$ and $z$ is 120.

(L. Šimünek)
Given values of the least common multiples will be decomposed into a product of prime numbers:

- $n(x, y)=200=2 \cdot 2 \cdot 2 \cdot 5 \cdot 5$,
- $n(y, z)=300=2 \cdot 2 \cdot 3 \cdot 5 \cdot 5$,
- $n(x, z)=120=2 \cdot 2 \cdot 2 \cdot 3 \cdot 5$.

We will sequentially record the prime factors of the decompositions of the numbers $x, y$, and $z$ in the table, adhering to the following principles:

- A prime number that is not in the decomposition of the least common multiple of two unknowns cannot be in the decompositions of these unknowns.
- How many times a certain prime number is in the decomposition of the least common multiple of two unknowns, that many times it must be in the decomposition of one of these unknowns and at most that many times it can be in the decomposition of the other unknown.

Since the prime number 2 is in the decomposition of $n(x, y)$ three times, it must be in the row of $x$ or in the row of $y$ three times. However, in the decomposition of $n(y, z)$, the prime number 2 is only twice, so it cannot be three times in the row of $y$. The prime number 2 is therefore three times in the row of $x$. Similarly, we will evaluate the occurrence of two prime numbers $5$ in the decomposition of $n(x, y)$ and one prime number 5 in the decomposition of $n(x, z)$, then the occurrence of the prime number 3 in the decomposition of $n(y, z)$ and its absence in the decomposition of $n(x, y)$. The table then looks like this:

| $x$ | $2 \cdot 2 \cdot 2 \ldots$ |
| :--- | :--- |
| $y$ | $5 \cdot 5 \ldots$ |
| $z$ | $3 \ldots$ |

If we continue to consider only the conditions about the least common multiples in the problem, we will not be able to add any prime number unambiguously to the table. We will therefore notice the condition that $x$ is the largest of the unknowns. In the row of $x$, we currently have a smaller value than in the row of $y$, so we need to add another factor to the row of $x$. The prime number 2 is already included in the maximum number, and the prime number 3 cannot be added because it is not in the decomposition of $n(x, y)$. We can only add the prime number 5, but only once, because it appears once in the decomposition of $n(x, z)$. We have thus determined the value of the first unknown:

| $x$ | $2 \cdot 2 \cdot 2 \cdot 5=40$ |
| :--- | :--- |
| $y$ | $5 \cdot 5 \ldots$ |
| $z$ | $3 \ldots$ |

No factor can be added to the row of $y$, otherwise $y$ would be greater than $x$. Therefore, $y=25$. To the row of $z$, we must then add two prime numbers 2 according to the decomposition of $n(y, z)$. The value in this row will then be 12, and no prime number 5 can be added because then the value in this row would be greater than in the row of $x$. The problem thus has a unique solution, as shown in the following table:

| $x$ | $2 \cdot 2 \cdot 2 \cdot 5=40$ |
| :--- | :--- |
| $y$ | $5 \cdot 5=25$ |
| $z$ | $2 \cdot 2 \cdot 3=12$ |

Evaluation. 2 points for determining that $x$ is divisible by eight, $y$ by twenty-five, and $z$ by three; 2 points for the final results; another 2 points based on the quality of the commentary.