Let's solve the problem step by step.

Let the positive integer be $n$, and let $q$ be the divisor.

1. The first condition is that when $n$ is divided by $q$, the remainder is 21: $n = kq + 21$ for some integer $k$.

2. The second condition is that when $2n$ is divided by $q$, the remainder is 13: $2n = mq + 13$ for some integer $m$.

Substituting $n = kq + 21$ into $2n$, we get: $2n = 2(kq + 21) = 2kq + 42$ When $2n$ is divided by $q$, the remainder is 13, so: $2kq + 42 \equiv 13 \pmod{q}$ This simplifies to: $42 \equiv 13 \pmod{q}$ Subtracting 13 from both sides: $29 \equiv 0 \pmod{q}$ This implies $q = 29$.

Thus, the value of $q$ is 29.

Would you like more details or have any questions?

Here are some related questions:

1. What is the general method for solving remainder problems in modular arithmetic?
2. Can you find other integer values for $q$ that satisfy similar conditions?
3. How can you determine the value of $n$ given $q$?
4. What is the significance of modulus in division problems?
5. How would the problem change if the remainder for $2n$ was larger than $q$?

Tip: When solving problems involving remainders, consider using modular arithmetic to simplify calculations and find patterns.