## Task 4 - 260914

Determine whether there exist natural numbers $n$ such that the solution $x$ of the equation $17 x + n = 6 x + 185$ is also a natural number! If this is the case, find the smallest such number $n$ and the corresponding solution $x$ of the given equation!
For every natural number $n$:

The equation $17 x+n=6 x+185$ can be equivalently transformed to $11 x=185-n$. It therefore has exactly the solution

$$
x=\frac{185-n}{11}
$$

This is a natural number for exactly those natural numbers $n$ for which $n \leq 185$ and 11 is a divisor of $185-n$.

These conditions are met, since 185 leaves a remainder of 9 when divided by 11, exactly by those natural numbers $n=9+11 m$ with integer $m$ for which $9+11 m \leq 185$ holds. Such natural numbers $n$ exist; the smallest of them is obtained with $m=0$.

The smallest number $n$ with the mentioned properties is thus $n=9$; according to (1), the corresponding solution of the equation given in the problem is the number $x=16$.

Solutions of the I. Round 1986 taken from [5]

### 6.28.2 II. Round 1986, Class 9