## Task 6 - 120836

Determine whether there is a smallest positive rational number $a$ for which there exists a natural number $x$ with the property

$$
\frac{25}{2} x - a = \frac{5}{8} x + 142
$$

If such a smallest $a$ exists, determine the value of $x$ for which this is true!
Assuming for a positive rational number $a$ there exists a natural number $x$ such that:

$$
\frac{25}{2} x - a = \frac{5}{8} x + 142
$$

Then it also holds that

$$
100 x - 8 a = 5 x + 1136 \quad \text{or} \quad 95 x = 1136 + 8 a, \quad \text{so} \quad x = \frac{1136 + 8 a}{95}
$$

Therefore, for a positive rational number $a$, there is a natural number $x$ with property (1) if and only if $1136 + 8 a$ is divisible by 95, where $8 a > 0$.

Since 1136 leaves a remainder of 91 when divided by 95, the smallest value of $a$ is obtained for $8 a = 4$, which is $a = \frac{1}{2}$.

For this value, equation (2) yields $x = 12$, which is a natural number. Conversely, $x = 12$ together with $a = \frac{1}{2}$ also satisfies (1), as can be verified by substitution. Thus, there is a smallest $a$ with the required properties, and $x$ takes the value 12 for this $a$.

Solutions of the III. Round 1972 taken from [5]

### 5.15 XIII. Olympiad 1973

### 5.15.1 I. Round 1973, Class 8