A positive integer n when divided by 9, gives 7 as remainder. What will be the remainder when (3n - 1) is divided by 9?

Options

A. 1
B. 2
C. 3
D. 4

Answer

To solve the problem step by step, we start with the information given:

1. **Understanding the problem**: We know that when a positive integer \( n \) is divided by 9, it gives a remainder of 7. This can be expressed mathematically as:
\[ n = 9k + 7 \]
where \( k \) is some integer.

2. **Finding \( 3n - 1 \)**: We need to find the expression for \( 3n - 1 \):
\[ 3n - 1 = 3(9k + 7) - 1 \]
Expanding this gives:
\[ 3n - 1 = 27k + 21 - 1 = 27k + 20 \]

3. **Dividing by 9**: Now, we want to find the remainder when \( 3n - 1 \) is divided by 9. We can rewrite \( 27k + 20 \) in a form that makes it easier to find the remainder:
\[ 27k + 20 = 9(3k) + 20 \]
Here, \( 9(3k) \) is clearly divisible by 9, so we only need to consider the term \( 20 \).

4. **Finding the remainder of 20 when divided by 9**: To find the remainder of \( 20 \) when divided by \( 9 \):
\[ 20 \div 9 = 2 \quad \text{(quotient)} \]
The product of the quotient and the divisor is:
\[ 9 \times 2 = 18 \]
Now, subtract this from \( 20 \):
\[ 20 - 18 = 2 \]
Thus, the remainder when \( 20 \) is divided by \( 9 \) is \( 2 \).

5. **Conclusion**: Therefore, the remainder when \( 3n - 1 \) is divided by \( 9 \) is:
\[ \boxed{2} \]