From the positive integers, $m,m+1,\dots,m+n$, only the sum of digits of $m$ and the sum of digits of $m+n$ are divisible by $8$. Find the maximum value of $n$.
1. **Identify the range of numbers:**
   We are given a sequence of positive integers starting from \( m \) to \( m+n \). We need to find the maximum value of \( n \) such that only the sum of the digits of \( m \) and \( m+n \) are divisible by 8.

2. **Consider the sum of digits modulo 8:**
   Let \( S(x) \) denote the sum of the digits of \( x \). We need \( S(m) \equiv 0 \pmod{8} \) and \( S(m+n) \equiv 0 \pmod{8} \), but for all \( k \) where \( 1 \leq k \leq n-1 \), \( S(m+k) \not\equiv 0 \pmod{8} \).

3. **Analyze the sequence in blocks of 10:**
   Consider a block of 10 consecutive numbers, \( 10a, 10a+1, \ldots, 10a+9 \). The sum of the digits of these numbers will vary, but at least one of them will have a sum of digits divisible by 8. This is because the sum of digits changes as we move from one number to the next.

4. **Extend the block to 18 numbers:**
   Similarly, consider the next block of 10 numbers, \( 10a+10, 10a+11, \ldots, 10a+19 \). Again, at least one of these numbers will have a sum of digits divisible by 8.

5. **Determine the maximum \( n \):**
   To ensure that only the first and the last numbers in the sequence have sums of digits divisible by 8, we need to find the maximum span where this condition holds. From the analysis, we see that within any block of 18 numbers, there will be at least two numbers whose sum of digits is divisible by 8.

6. **Calculate the maximum \( n \):**
   If \( m \) is such that \( S(m) \equiv 0 \pmod{8} \) and \( S(m+n) \equiv 0 \pmod{8} \), the maximum \( n \) can be \( 17 \) because within any block of 18 numbers, there will be at least two numbers with sums of digits divisible by 8. Therefore, the maximum \( n \) is \( 17 - 2 = 15 \).

7. **Verify with an example:**
   For example, if \( m = 9999992 \), then \( S(9999992) = 9+9+9+9+9+9+2 = 56 \equiv 0 \pmod{8} \). Similarly, \( S(9999992 + 15) = S(10000007) = 1+0+0+0+0+0+7 = 8 \equiv 0 \pmod{8} \). All other sums of digits in between are not divisible by 8.

\[
\boxed{15}
\]