Find all positive integers $n$ that can be subtracted from both the numerator and denominator of the fraction $\frac{1234}{6789}$, to get, after the reduction, the fraction of form $\frac{a}{b}$, where $a, b$ are single digit numbers. 

[i]Proposed by Bogdan Rublov[/i]
1. We start with the fraction $\frac{1234}{6789}$ and we want to find all positive integers $n$ such that $\frac{1234 - n}{6789 - n}$ reduces to a fraction $\frac{a}{b}$ where $a$ and $b$ are single-digit numbers.

2. Since $a$ and $b$ are single-digit numbers, the fraction $\frac{a}{b}$ must be one of the following: $\frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}, \frac{1}{6}, \frac{1}{7}, \frac{1}{8}, \frac{1}{9}, \frac{2}{3}, \frac{2}{5}, \frac{3}{4}$, etc.

3. We need to find $n$ such that:
   \[
   \frac{1234 - n}{6789 - n} = \frac{a}{b}
   \]
   This can be rewritten as:
   \[
   b(1234 - n) = a(6789 - n)
   \]
   Simplifying, we get:
   \[
   b \cdot 1234 - b \cdot n = a \cdot 6789 - a \cdot n
   \]
   \[
   b \cdot 1234 - a \cdot 6789 = n(b - a)
   \]
   \[
   n = \frac{b \cdot 1234 - a \cdot 6789}{b - a}
   \]

4. We need to check which values of $a$ and $b$ yield an integer $n$ that is positive and makes $a$ and $b$ single-digit numbers.

5. Let's start with the smallest possible fraction $\frac{1}{b}$:
   \[
   \frac{1}{b} = \frac{1234 - n}{6789 - n}
   \]
   This implies:
   \[
   6789 - n = b(1234 - n)
   \]
   \[
   6789 - n = 1234b - bn
   \]
   \[
   6789 - 1234b = n(b - 1)
   \]
   \[
   n = \frac{6789 - 1234b}{b - 1}
   \]

6. We need to check for which $b$ this $n$ is a positive integer:
   - For $b = 2$:
     \[
     n = \frac{6789 - 1234 \cdot 2}{2 - 1} = \frac{6789 - 2468}{1} = 4321 \quad (\text{not a single-digit fraction})
     \]
   - For $b = 3$:
     \[
     n = \frac{6789 - 1234 \cdot 3}{3 - 1} = \frac{6789 - 3702}{2} = 1543.5 \quad (\text{not an integer})
     \]
   - For $b = 4$:
     \[
     n = \frac{6789 - 1234 \cdot 4}{4 - 1} = \frac{6789 - 4936}{3} = 617.67 \quad (\text{not an integer})
     \]
   - For $b = 5$:
     \[
     n = \frac{6789 - 1234 \cdot 5}{5 - 1} = \frac{6789 - 6170}{4} = 154.75 \quad (\text{not an integer})
     \]
   - For $b = 6$:
     \[
     n = \frac{6789 - 1234 \cdot 6}{6 - 1} = \frac{6789 - 7404}{5} = -123 \quad (\text{not positive})
     \]
   - For $b = 7$:
     \[
     n = \frac{6789 - 1234 \cdot 7}{7 - 1} = \frac{6789 - 8638}{6} = -308.17 \quad (\text{not positive})
     \]
   - For $b = 8$:
     \[
     n = \frac{6789 - 1234 \cdot 8}{8 - 1} = \frac{6789 - 9872}{7} = -370.43 \quad (\text{not positive})
     \]
   - For $b = 9$:
     \[
     n = \frac{6789 - 1234 \cdot 9}{9 - 1} = \frac{6789 - 11106}{8} = -531.75 \quad (\text{not positive})
     \]

7. We see that none of the values of $b$ from 2 to 9 yield a positive integer $n$.

8. Therefore, there are no positive integers $n$ that can be subtracted from both the numerator and denominator of the fraction $\frac{1234}{6789}$ to get a fraction of the form $\frac{a}{b}$ where $a$ and $b$ are single-digit numbers.

The final answer is $\boxed{None}$