If 7 divided a positive integer n, the remainder is 2. Which of the following numbers gives a remainder of 0 when divided by 7?

Options

A. n +2
B. n +5
C. n - 5
D. n +1

Answer

To solve the problem, we need to find which of the given expressions will yield a remainder of 0 when divided by 7, given that \( n \) leaves a remainder of 2 when divided by 7.

### Step-by-Step Solution:

1. **Understanding the Problem:**
We know that when a positive integer \( n \) is divided by 7, it leaves a remainder of 2. This can be expressed mathematically as:
\[ n = 7k + 2 \]
where \( k \) is some integer (the quotient).

2. **Identifying the Expressions:**
We need to check the following expressions:
- \( n + 2 \)
- \( n + 5 \)
- \( n - 5 \)
- \( n + 1 \)

3. **Substituting \( n \) into Each Expression:**
Let's substitute \( n = 7k + 2 \) into each expression and simplify.

- For \( n + 2 \):
\[ n + 2 = (7k + 2) + 2 = 7k + 4 \]
When divided by 7, the remainder is 4.

- For \( n + 5 \):
\[ n + 5 = (7k + 2) + 5 = 7k + 7 \]
When divided by 7, the remainder is 0.

- For \( n - 5 \):
\[ n - 5 = (7k + 2) - 5 = 7k - 3 \]
When divided by 7, the remainder is 4 (since -3 is equivalent to 4 mod 7).

- For \( n + 1 \):
\[ n + 1 = (7k + 2) + 1 = 7k + 3 \]
When divided by 7, the remainder is 3.

4. **Conclusion:**
The only expression that gives a remainder of 0 when divided by 7 is \( n + 5 \).

### Final Answer:
The correct option is \( n + 5 \).