## Task 4 - 300624

Determine all natural numbers $n$ that can be represented in the form $n=5a+7b$, where $a$ and $b$ are natural numbers!
The following table shows all values $\mathrm{n}=5 \mathrm{a}+7 \mathrm{~b}$ with $\mathrm{a}=0,1,2,3,4,5$ and $\mathrm{b}=0,1,2,3,4$

| $\mathrm{b} \mid \mathrm{a}$ | 0 | 1 | 2 | 3 | 4 | 5 |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| 0 | 0 | 5 | 10 | 15 | 20 | 25 |
| 1 | 7 | 12 | 17 | 22 | 27 | 32 |
| 2 | 14 | 19 | 24 | 29 | 34 | 39 |
| 3 | 21 | 26 | 31 | 36 | 41 | 46 |
| 4 | 28 | 33 | 38 | 43 | 48 | 53 |

Since increasing $a$ or $b$ (or both) always results in a larger $n$, it follows that:

(1) Among all natural numbers $n \leq 24$, exactly the numbers $0,5,7,10,12,14,15,17,19,20$, 21, 22, 24 can be represented in the specified form. Furthermore, it is clear from the table that:

(2) The numbers 24, 25, 26, 27, 28 can be represented in the specified form. By adding $1 \cdot 5, 2 \cdot 5, 3 \cdot 5, \ldots$ and so on to the numbers mentioned in (2), we obtain:

(3) The numbers 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43 can also be represented in the specified form.

With (2) and (3), it is shown that every natural number $n \geq 24$ can be represented in this form. The sought numbers are therefore exactly the numbers mentioned in (1) and all natural numbers $n>24$.

Solutions of the II. Round 1990 taken from [5]

### 3.32 XXXI. Olympiad 1991

### 3.32.1 I. Round 1991, Class 6