## Task 4 - 190824

Klaus says:

"I am thinking of three natural numbers. The second number is 2 more than half of the first number. The third number is 2 more than half of the second number. The product of the three thought-of numbers is 1120.

Which number did I think of first, which as the second, and which as the third?"

Can this question be answered unambiguously? If so, name the three thought-of numbers!
I) If three numbers have the mentioned properties and \( n \) is the first number, then the second number is \(\frac{n}{2} + 2 = \frac{n + 4}{2}\) and the third number is \(\frac{n + 4}{4} + 2 = \frac{n}{4} + 3\). Since this is also a natural number, \( n \) is divisible by 4.

If \( n \) were a (divisible by 4) natural number with \( n \leq 12 \), then \(\frac{n}{2} + 2\) and \(\frac{n}{4} + 3\) would be natural numbers with \(\frac{n}{2} + 2 > 12\) and \(\frac{n}{4} + 3 > 8\), thus the product \(\geq 1920\), and therefore greater than 1120.

Thus, the only possible first number is \( n = 16 \), the second number is \(\frac{n}{2} + 2 = 10\), and the third number is \(\frac{n}{4} + 3 = 7\).

II) These numbers have the required properties as shown by the verification.

Therefore, Klaus's question can be answered uniquely. He thought of 16 as the first number, 10 as the second, and 7 as the third.

Solutions of the II. Round 1979 adopted from [5]

### 5.21.3 III. Round 1979, Class 8