How many pairs of A and B are possible in number `89765A4B`, If number is divisible by `18?`

Options

A. 4
B. 5
C. 6
D. 7

Answer

To determine how many pairs of A and B are possible in the number `89765A4B` such that the number is divisible by `18`, we need to check the conditions for divisibility by both `2` and `9`.

### Step 1: Check divisibility by 2
A number is divisible by `2` if its last digit (unit place) is even. In our case, the last digit is represented by B. Therefore, B must be one of the following even digits: `0, 2, 4, 6, 8`.

**Hint:** List all even digits that B can take.

### Step 2: Check divisibility by 9
A number is divisible by `9` if the sum of its digits is divisible by `9`. We first calculate the sum of the known digits in `89765A4B`:

- Sum of known digits:
- \(8 + 9 + 7 + 6 + 5 + 4 = 39\)

Now, including A and B, the total sum becomes:
- Total sum = \(39 + A + B\)

For the number to be divisible by `9`, \(39 + A + B\) must be divisible by `9`.

**Hint:** Write the equation for the sum of the digits and express the condition for divisibility by 9.

### Step 3: Analyze each possible value of B
Now we will analyze each possible value of B (0, 2, 4, 6, 8) and determine the corresponding values of A that make \(39 + A + B\) divisible by `9`.

1. **If B = 0:**
- \(39 + A + 0 = 39 + A\)
- \(39 + A\) must be divisible by `9`.
- Possible values of A: \(A = 0, 6\) (since \(39 + 0 = 39\) and \(39 + 6 = 45\))

2. **If B = 2:**
- \(39 + A + 2 = 41 + A\)
- Possible values of A: \(A = 4\) (since \(41 + 4 = 45\))

3. **If B = 4:**
- \(39 + A + 4 = 43 + A\)
- Possible values of A: \(A = 2, 9\) (since \(43 + 2 = 45\) and \(43 + 9 = 52\))

4. **If B = 6:**
- \(39 + A + 6 = 45 + A\)
- Possible value of A: \(A = 0\) (since \(45 + 0 = 45\))

5. **If B = 8:**
- \(39 + A + 8 = 47 + A\)
- Possible value of A: \(A = 7\) (since \(47 + 7 = 54\))

### Step 4: Compile all possible pairs (A, B)
Now we compile all the pairs (A, B) that satisfy both conditions:

- For B = 0: (0, 0), (6, 0)
- For B = 2: (4, 2)
- For B = 4: (2, 4), (9, 4)
- For B = 6: (0, 6)
- For B = 8: (7, 8)

### Step 5: Count the total pairs
Counting all the valid pairs, we have:
- From B = 0: 2 pairs
- From B = 2: 1 pair
- From B = 4: 2 pairs
- From B = 6: 1 pair
- From B = 8: 1 pair

Total pairs = \(2 + 1 + 2 + 1 + 1 = 7\)

### Conclusion
The total number of pairs (A, B) such that the number `89765A4B` is divisible by `18` is **7**.

**Final Answer:** 7

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