On dividing a positive integers `n` by `9`, we get `7` as remainder. What will be the remainder if `( 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 \( 9 \), the remainder is \( 7 \). This can be mathematically expressed as:
\[ n \equiv 7 \ (\text{mod} \ 9) \]

2. **Expressing \( n \)**: From the above congruence, we can express \( n \) in terms of \( k \) (where \( k \) is some integer):
\[ n = 9k + 7 \]

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

4. **Finding the Remainder when Divided by 9**: Now, we need to find the remainder when \( 27k + 20 \) is divided by \( 9 \):
- Since \( 27k \) is divisible by \( 9 \) (as \( 27 = 3 \times 9 \)), it contributes \( 0 \) to the remainder.
- Now we only need to consider \( 20 \):
\[ 20 \div 9 = 2 \quad \text{(which gives a quotient of 2 and a remainder of 2)} \]
Therefore, the remainder when \( 20 \) is divided by \( 9 \) is:
\[ 20 \equiv 2 \ (\text{mod} \ 9) \]

5. **Final Answer**: Thus, the remainder when \( 3n - 1 \) is divided by \( 9 \) is:
\[ \text{Remainder} = 2 \]