## Task 3 - 110913

Determine all natural numbers $a$ for which the term

$$
t=\frac{a+11}{a-9}
$$

is a natural number!
For natural numbers $a9$ and setting $h=a-9$, then $h$ is always a natural number, further $a=h+9$ and

$$
t=\frac{h+20}{h}=1+\frac{20}{h}
$$

Thus, $t$ is a natural number if and only if $h$ is a divisor of 20. Therefore, the following possibilities arise:

| $h$ | $a=h+9$ | $t$ |
| ---: | :---: | ---: |
| 1 | 10 | 21 |
| 2 | 11 | 11 |
| 4 | 13 | 6 |
| 5 | 14 | 5 |
| 10 | 19 | 3 |
| 20 | 29 | 2 |

Hence, exactly the numbers 10, 11, 13, 14, 19, and 20 satisfy all the given conditions.

Taken from $[2]$