A and B are positive integers. If A + B + AB = 65, then what is the difference between A and B (A, B `le` 15) ?

Options

A. 3
B. 4
C. 5
D. 6

Answer

To solve the problem, we need to find two positive integers A and B such that:

\[ A + B + AB = 65 \]

and both \( A \) and \( B \) are less than or equal to 15.

### Step-by-Step Solution:

1. **Rewrite the equation**:
We start with the equation:
\[ A + B + AB = 65 \]
We can rearrange it as:
\[ AB + A + B = 65 \]

2. **Introduce a new variable**:
We can add 1 to both sides to factor the left-hand side:
\[ AB + A + B + 1 = 66 \]
This can be factored as:
\[ (A + 1)(B + 1) = 66 \]

3. **Find factor pairs of 66**:
Next, we need to find pairs of factors of 66. The factor pairs of 66 are:
- (1, 66)
- (2, 33)
- (3, 22)
- (6, 11)

4. **Convert factor pairs back to A and B**:
For each factor pair \( (x, y) \), we can set:
\[ A + 1 = x \quad \text{and} \quad B + 1 = y \]
or
\[ A + 1 = y \quad \text{and} \quad B + 1 = x \]
Then we can find \( A \) and \( B \) by subtracting 1 from each factor.

5. **Check valid pairs**:
We will check each factor pair to see if they yield valid integers for \( A \) and \( B \) that are less than or equal to 15:
- For \( (6, 11) \):
- \( A + 1 = 6 \) and \( B + 1 = 11 \) gives \( A = 5 \) and \( B = 10 \) (valid)
- \( A + 1 = 11 \) and \( B + 1 = 6 \) gives \( A = 10 \) and \( B = 5 \) (valid)

6. **Calculate the difference**:
Now we can find the difference between \( A \) and \( B \):
\[ |A - B| = |10 - 5| = 5 \]

### Final Answer:
The difference between \( A \) and \( B \) is \( 5 \).