When an integer n is divided by 7, the remainder is 3. What will be the remainder if 6n is divided by 7 ?

Options

A. 2
B. 4
C. 1
D. 0

Answer

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

1. When an integer \( n \) is divided by 7, the remainder is 3. This can be expressed mathematically as:
\[ n = 7k + 3 \]
where \( k \) is some integer.

2. We need to find the remainder when \( 6n \) is divided by 7. First, we calculate \( 6n \):
\[ 6n = 6(7k + 3) = 42k + 18 \]

3. Now, we need to find the remainder of \( 42k + 18 \) when divided by 7. We can simplify this by finding the remainders of each term:
- The term \( 42k \) is divisible by 7, so:
\[ 42k \mod 7 = 0 \]
- Next, we find the remainder of 18 when divided by 7:
\[ 18 \div 7 = 2 \quad \text{(since } 7 \times 2 = 14\text{)} \]
The remainder is:
\[ 18 - 14 = 4 \]

4. Therefore, when we combine these results, we have:
\[ 6n \mod 7 = (42k + 18) \mod 7 = (0 + 4) \mod 7 = 4 \]

Thus, the remainder when \( 6n \) is divided by 7 is **4**.