If 5 divides the integer n, the remainder is 2. What will be the remainder if 7n is divided by 5 ?

Options

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

Answer

To solve the problem, we need to find the remainder when \( 7n \) is divided by 5, given that \( n \) is an integer such that when \( n \) is divided by 5, the remainder is 2.

### Step-by-step Solution:

1. **Understanding the relationship of \( n \) with 5**:
Since \( n \) gives a remainder of 2 when divided by 5, we can express \( n \) in the following form:
\[ n = 5k + 2 \]
where \( k \) is some integer.

**Hint**: Remember that any integer can be expressed in terms of its divisor and remainder.

2. **Calculating \( 7n \)**:
Now, we will multiply \( n \) by 7:
\[ 7n = 7(5k + 2) = 35k + 14 \]

**Hint**: Distributing the multiplication over addition can help simplify expressions.

3. **Finding the remainder of \( 7n \) when divided by 5**:
We need to divide \( 7n \) by 5 and find the remainder:
\[ 7n = 35k + 14 \]
When we divide \( 35k \) by 5, the term \( 35k \) is divisible by 5, so it gives a remainder of 0.

Now we need to find the remainder of \( 14 \) when divided by 5:
\[ 14 \div 5 = 2 \quad \text{(quotient)} \]
\[ 14 - (5 \times 2) = 14 - 10 = 4 \quad \text{(remainder)} \]

**Hint**: To find the remainder, subtract the largest multiple of 5 that is less than or equal to the number.

4. **Conclusion**:
Therefore, the remainder when \( 7n \) is divided by 5 is:
\[ \text{Remainder} = 4 \]

**Final Answer**: The remainder is 4.