We are given that $n$ is a positive number such that when $n$ is divided by 7, the remainder is 2. In mathematical terms:

$n \equiv 2 \pmod{7}.$

We want to find the remainder when $9n$ is divided by 7. Using modular arithmetic, we calculate:

1. Start with the given congruence: $n \equiv 2 \pmod{7}.$

2. Multiply through by 9: $9n \equiv 9 \cdot 2 \pmod{7}.$

3. Simplify $9 \cdot 2$: $9 \cdot 2 = 18.$

4. Find $18 \mod 7$: $18 \div 7 = 2 \text{ remainder } 4,$ so: $18 \equiv 4 \pmod{7}.$

5. Conclude: $9n \equiv 4 \pmod{7}.$

Thus, the remainder when $9n$ is divided by 7 is 4.

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Related Questions:

1. What is modular arithmetic, and how is it applied in real-world scenarios?
2. How do you compute the remainder of a product in modular arithmetic?
3. Can you explain the concept of congruences in number theory?
4. What is the relationship between modular arithmetic and division with remainders?
5. How does multiplying congruences affect their remainders?

Tip:

When working with modular arithmetic, always simplify the numbers by reducing them modulo the divisor before performing operations. This keeps calculations manageable.