If 24AB4 is divisible by 99, then `A xx B` is :

Options

A. 25
B. 30
C. 20
D. 15

Answer

To determine the values of A and B such that the number 24AB4 is divisible by 99, we need to check for divisibility by both 9 and 11, since 99 = 9 × 11.

### Step-by-step Solution:

**Step 1: Check for divisibility by 9**
To check if a number is divisible by 9, the sum of its digits must be divisible by 9.

The digits of the number 24AB4 are:
- 2
- 4
- A
- B
- 4

The sum of these digits is:
\[ 2 + 4 + A + B + 4 = 10 + A + B \]

For divisibility by 9:
\[ 10 + A + B \equiv 0 \mod 9 \]

This means:
\[ A + B \equiv -10 \equiv -1 \equiv 8 \mod 9 \]
Thus, we have:
\[ A + B = 8 \quad \text{or} \quad A + B = 17 \]

**Step 2: Check for divisibility by 11**
To check if a number is divisible by 11, the difference between the sum of the digits in odd positions and the sum of the digits in even positions must be divisible by 11.

The odd-positioned digits are:
- 2 (1st position)
- A (3rd position)
- 4 (5th position)

The even-positioned digits are:
- 4 (2nd position)
- B (4th position)

Calculating the sums:
- Sum of digits in odd positions: \( 2 + A + 4 = A + 6 \)
- Sum of digits in even positions: \( 4 + B \)

The difference is:
\[ |(A + 6) - (4 + B)| = |A - B + 2| \]

For divisibility by 11:
\[ |A - B + 2| \equiv 0 \mod 11 \]

This means:
\[ A - B + 2 = 0 \quad \text{or} \quad A - B + 2 = 11 \]
Thus, we have:
1. \( A - B = -2 \) (which simplifies to \( A = B - 2 \))
2. \( A - B = 9 \) (which simplifies to \( A = B + 9 \))

**Step 3: Solve the equations**
We will use the equations from steps 1 and 2.

From \( A + B = 8 \) and \( A = B - 2 \):
Substituting \( A \) in the first equation:
\[ (B - 2) + B = 8 \]
\[ 2B - 2 = 8 \]
\[ 2B = 10 \]
\[ B = 5 \]

Now substituting \( B \) back to find \( A \):
\[ A + 5 = 8 \]
\[ A = 3 \]

**Step 4: Calculate A × B**
Now that we have \( A = 3 \) and \( B = 5 \):
\[ A \times B = 3 \times 5 = 15 \]

### Final Answer:
Thus, the value of \( A \times B \) is **15**.