How many numbers are there that appear both in the arithmetic sequence $10,
16, 22, 28, ... 1000$ and the arithmetic sequence $10, 21, 32, 43, ..., 1000?$
1. **Identify the general terms of both sequences:**
   - The first arithmetic sequence is \(10, 16, 22, 28, \ldots, 1000\). The first term \(a_1 = 10\) and the common difference \(d_1 = 6\). The general term of this sequence can be written as:
     \[
     a_n = 10 + (n-1) \cdot 6 = 6n + 4
     \]
   - The second arithmetic sequence is \(10, 21, 32, 43, \ldots, 1000\). The first term \(b_1 = 10\) and the common difference \(d_2 = 11\). The general term of this sequence can be written as:
     \[
     b_m = 10 + (m-1) \cdot 11 = 11m - 1
     \]

2. **Find the common terms in both sequences:**
   - We need to find the values of \(n\) and \(m\) such that:
     \[
     6n + 4 = 11m - 1
     \]
   - Rearrange the equation:
     \[
     6n - 11m = -5
     \]

3. **Solve the Diophantine equation \(6n - 11m = -5\):**
   - This is a linear Diophantine equation. We need to find integer solutions for \(n\) and \(m\). First, find a particular solution. By inspection, \(n = 1\) and \(m = 1\) is a solution:
     \[
     6(1) - 11(1) = -5
     \]
   - The general solution to the equation \(6n - 11m = -5\) can be written as:
     \[
     n = 1 + 11k \quad \text{and} \quad m = 1 + 6k \quad \text{for integer } k
     \]

4. **Determine the range of \(k\) such that the terms are within the bounds of the sequences:**
   - The terms must be between 10 and 1000. For the first sequence:
     \[
     10 \leq 6n + 4 \leq 1000
     \]
     \[
     6 \leq 6n \leq 996
     \]
     \[
     1 \leq n \leq 166
     \]
   - For the second sequence:
     \[
     10 \leq 11m - 1 \leq 1000
     \]
     \[
     11 \leq 11m \leq 1001
     \]
     \[
     1 \leq m \leq 91
     \]
   - Substitute \(n = 1 + 11k\) and \(m = 1 + 6k\) into the ranges:
     \[
     1 \leq 1 + 11k \leq 166 \quad \Rightarrow \quad 0 \leq 11k \leq 165 \quad \Rightarrow \quad 0 \leq k \leq 15
     \]
     \[
     1 \leq 1 + 6k \leq 91 \quad \Rightarrow \quad 0 \leq 6k \leq 90 \quad \Rightarrow \quad 0 \leq k \leq 15
     \]

5. **Count the number of valid \(k\) values:**
   - The integer values of \(k\) range from 0 to 15, inclusive. Therefore, there are \(15 - 0 + 1 = 16\) valid values of \(k\).

The final answer is \(\boxed{16}\)