Use the method of plugging In to solve the following problem.
The number m yields a remainder p when divided by 14 and a remainder q when divided by 7. If `p = q + 7`, then which of the following could be the value of m?

Options

A. 45
B. 53
C. 72
D. 81

Answer

To solve the problem using the method of plugging in, we need to analyze the conditions given in the question.

### Step-by-Step Solution:

1. **Understand the problem**: We are given that a number \( m \) yields a remainder \( p \) when divided by 14 and a remainder \( q \) when divided by 7. It is also given that \( p = q + 7 \).

2. **Set up the equations**:
- From the first condition, we can express \( m \) in terms of \( p \):
\[ m = 14k + p \quad \text{(for some integer } k\text{)} \]
- From the second condition, we express \( m \) in terms of \( q \):
\[ m = 7j + q \quad \text{(for some integer } j\text{)} \]

3. **Substituting \( p \)**: Since \( p = q + 7 \), we can substitute \( p \) in the first equation:
\[ m = 14k + (q + 7) = 14k + q + 7 \]

4. **Equating the two expressions for \( m \)**:
\[ 14k + q + 7 = 7j + q \]
- Simplifying this, we get:
\[ 14k + 7 = 7j \]
- Dividing the entire equation by 7:
\[ 2k + 1 = j \]

5. **Finding possible values of \( m \)**: Now we can express \( m \) in terms of \( k \):
\[ m = 14k + (q + 7) \]
- Since \( q \) can take values from 0 to 6 (as it is a remainder when divided by 7), we can plug in values of \( k \) and \( q \) to find possible values of \( m \).

6. **Testing the options**:
- **Option A: \( m = 45 \)**
- \( 45 \div 14 = 3 \) remainder \( 3 \) (so \( p = 3 \))
- \( 45 \div 7 = 6 \) remainder \( 3 \) (so \( q = 3 \))
- Here, \( p \neq q + 7 \) (since \( 3 \neq 3 + 7 \)).

- **Option B: \( m = 53 \)**
- \( 53 \div 14 = 3 \) remainder \( 11 \) (so \( p = 11 \))
- \( 53 \div 7 = 7 \) remainder \( 4 \) (so \( q = 4 \))
- Here, \( p = q + 7 \) (since \( 11 = 4 + 7 \)).

- **Option C: \( m = 72 \)**
- \( 72 \div 14 = 5 \) remainder \( 2 \) (so \( p = 2 \))
- \( 72 \div 7 = 10 \) remainder \( 2 \) (so \( q = 2 \))
- Here, \( p \neq q + 7 \) (since \( 2 \neq 2 + 7 \)).

- **Option D: \( m = 81 \)**
- \( 81 \div 14 = 5 \) remainder \( 11 \) (so \( p = 11 \))
- \( 81 \div 7 = 11 \) remainder \( 4 \) (so \( q = 4 \))
- Here, \( p = q + 7 \) (since \( 11 = 4 + 7 \)).

7. **Conclusion**: The values of \( m \) that satisfy the condition \( p = q + 7 \) are:
- **Option B: 53**
- **Option D: 81**