Let's clarify how we arrived at 42 in the solution.

Given:
1. \( n = kq + 21 \)
2. When doubled, \( 2n \) leaves a remainder of 13 when divided by \( q \).

### Step-by-Step Explanation:

1. **Expression for \( n \):**  
   \( n = kq + 21 \), where \( k \) is an integer and 21 is the remainder.

2. **Double the number:**  
   Doubling \( n \) gives:
   \[
   2n = 2(kq + 21) = 2kq + 42
   \]
   Here, \( 42 \) comes from multiplying the remainder 21 by 2:
   \[
   2 \times 21 = 42
   \]

3. **Modulus operation:**  
   We are given that when \( 2n \) is divided by \( q \), the remainder is 13:
   \[
   2kq + 42 \equiv 13 \pmod{q}
   \]
   Since \( 2kq \) is divisible by \( q \) (it leaves no remainder), the remainder must come from 42:
   \[
   42 \equiv 13 \pmod{q}
   \]

4. **Simplifying to find \( q \):**  
   Subtract 13 from both sides:
   \[
   29 \equiv 0 \pmod{q}
   \]
   This tells us that \( q \) must divide 29 exactly, so \( q = 29 \).

In summary, the 42 was derived from multiplying the original remainder (21) by 2 when calculating \( 2n \).

Do you need any further details or have any more questions?

Here are some related questions:
1. What happens if the original remainder is larger than \( q \)?
2. How would the process change if tripling \( n \) instead of doubling it?
3. Can you generalize this approach to find \( q \) for any remainder combination?
4. How does the solution change if the remainder after doubling was 0?
5. What other methods could verify the solution for \( q \)?

**Tip:** Always carefully track operations like multiplication when working with remainders, as they directly affect the calculation of subsequent remainders.

Explore the solution to a mathematical problem involving modular arithmetic and remainders. Learn how doubling the original remainder (21) results in 42, and how this relates to finding the value of q. Suitable for advanced mathematics enthusiasts.

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