## Task 4 - 220724

For three natural numbers $a, b, c$, the following properties (1) and (2) are required:

(1) It holds that $a<b<c$.

(2) If $a, b, c$ are the measurements in centimeters of the edge lengths of a cuboid, then the cuboid has a volume of $270 \mathrm{~cm}^{3}$, and the sum of the lengths of all twelve edges of the cuboid is $80 \mathrm{~cm}$.

Investigate whether there are natural numbers that satisfy these requirements, and whether these numbers are uniquely determined by the requirements (1) and (2)! If this is the case, name these numbers!
Natural numbers $a, b, c$ satisfy condition (2) if and only if they satisfy the equations

(3) $a b c = 270$,

(4) $a + b + c = 20$.

I. If natural numbers $a, b, c$ satisfy conditions (1), (3), (4), then:

From (3), $a, b, c$ are different from 0; hence, and due to (1), (4), it follows that (5) $0 < a < b < c < 20$.

The only divisors of 270 between 0 and 20 are (6) 1, 2, 3, 5, 6, 9, 10, 15, and 18.

The only possibilities to choose two of these numbers as $a$ and $b$ with $a < b$ such that the number $c = 20 - a - b$ also satisfies $b < c$ are listed in the following table.

For those $a, b$ for which this number $c = 20 - a - b$ is also one of the numbers (6), it is then checked whether $a b c = 270$ also holds:

| $a$ | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 5 |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| $b$ | 2 | 3 | 5 | 6 | 9 | 3 | 5 | 6 | 5 | 6 | 6 |
| $c$ | 17 | 16 | 14 | 13 | 10 | 15 | 13 | 12 | 12 | 11 | 9 |
| $c$ in (6)? | no | no | no | no | yes | yes | no | no | no | no | yes |
| $a b c$ |  |  |  |  | 90 | 90 |  |  |  |  | 270 |

It follows that only $a = 5, b = 6, c = 9$ can satisfy conditions (1), (3), (4).

II. These numbers satisfy these conditions; indeed, $5 < 6 < 9, 5 \cdot 6 \cdot 9 = 270, 5 + 6 + 9 = 20$.

Thus, it is shown: There are numbers that satisfy the requirements (1), (2), they are uniquely determined by these requirements and are $a = 5, b = 6, c = 9$.

Solutions of the second round 1982 taken from [5]

### 4.24.3 Third Round 1982, Class 7