The digits of a three-digit number A are written in the reverse order to form another three-digit number B. If `BgtAandB-A` is perfectly divisible by 7, then which of the following is necessarily true?

Options

A. `100ltAlt299`
B. `106ltAlt305`
C. `112lt4lt311`
D. `118ltAlt317`

Answer

To solve the problem step by step, we start by defining the three-digit number \( A \) and its reverse \( B \).

### Step 1: Define the Numbers
Let \( A \) be represented as:
\[ A = 100k + 10l + m \]
where \( k, l, m \) are the digits of \( A \) (with \( k \) being the hundreds digit, \( l \) the tens digit, and \( m \) the units digit).

The reverse of \( A \), which is \( B \), can be represented as:
\[ B = 100m + 10l + k \]

### Step 2: Calculate \( B - A \)
Now, we calculate \( B - A \):
\[ B - A = (100m + 10l + k) - (100k + 10l + m) \]
Simplifying this gives:
\[ B - A = 100m + 10l + k - 100k - 10l - m \]
\[ B - A = 100m - 100k + k - m \]
\[ B - A = 99m - 99k \]
\[ B - A = 99(m - k) \]

### Step 3: Divisibility Condition
We know from the problem statement that \( B - A \) is perfectly divisible by 7:
\[ 99(m - k) \text{ is divisible by } 7 \]
Since \( 99 \) is not divisible by \( 7 \), it follows that \( m - k \) must be divisible by \( 7 \).

### Step 4: Express \( m \) in terms of \( k \)
From the divisibility condition, we can express:
\[ m - k = 7n \quad \text{for some integer } n \]
Thus, we can write:
\[ m = 7n + k \]

### Step 5: Determine the Range of Values
Since \( m \) is a digit (0 to 9), we need to find valid values for \( n \) such that \( m \) remains a single digit:
- If \( n = 0 \), \( m = k \)
- If \( n = 1 \), \( m = 7 + k \)

For \( n = 1 \), \( k \) must be \( 0 \) or \( 1 \) (since \( m \) must be less than or equal to 9):
- If \( k = 0 \), \( m = 7 \)
- If \( k = 1 \), \( m = 8 \)
- If \( k = 2 \), \( m = 9 \)

If \( k \) were to be \( 3 \) or higher, \( m \) would exceed \( 9 \).

### Step 6: Minimum and Maximum Values of \( A \)
Now, we can find the minimum and maximum values of \( A \):
- **Minimum Value of \( A \)**:
- \( k = 1, l = 0, m = 8 \) gives \( A = 100(1) + 10(0) + 8 = 108 \).

- **Maximum Value of \( A \)**:
- \( k = 2, l = 9, m = 9 \) gives \( A = 100(2) + 10(9) + 9 = 299 \).

### Conclusion
Thus, \( A \) can take values from \( 108 \) to \( 299 \).

### Final Answer
The only option that fits within this range is:
- \( A \) is greater than \( 106 \) and less than \( 305 \).