Joe was asked to fill up the missing digits A and B in the number 25A7B subject to the condition that the resulting number has to be divisible by 36.What is the value of (A + B) if A and B are distinct digits ?

Options

A. 4
B. 8
C. 9
D. 13

Answer

To solve the problem of finding the missing digits A and B in the number 25A7B such that the number is divisible by 36, we need to follow these steps:

### Step 1: Understand the conditions for divisibility by 36
A number is divisible by 36 if it is divisible by both 4 and 9.

### Step 2: Check divisibility by 4
For a number to be divisible by 4, the last two digits must form a number that is divisible by 4. In our case, the last two digits are 7B.

- The possible values for B (0-9) will give us the following two-digit numbers:
- 70, 71, 72, 73, 74, 75, 76, 77, 78, 79.

- Among these, the numbers divisible by 4 are:
- 72 (7B = 72, B = 2)
- 76 (7B = 76, B = 6)

Thus, B can be either 2 or 6.

### Step 3: Check divisibility by 9
For a number to be divisible by 9, the sum of its digits must be divisible by 9. The sum of the digits in 25A7B is:
\[ 2 + 5 + A + 7 + B = 14 + A + B \]

We need to check the cases for B = 2 and B = 6.

### Case 1: B = 2
Substituting B = 2 into the sum:
\[ 14 + A + 2 = 16 + A \]
We need this to be divisible by 9. The possible values of \( 16 + A \) that are divisible by 9 are:
- 18 (which gives A = 2)
- 27 (which gives A = 11, not possible since A must be a single digit)

Since A = 2 is not distinct from B, we discard this case.

### Case 2: B = 6
Substituting B = 6 into the sum:
\[ 14 + A + 6 = 20 + A \]
We need this to be divisible by 9. The possible values of \( 20 + A \) that are divisible by 9 are:
- 27 (which gives A = 7)
- 36 (which gives A = 16, not possible since A must be a single digit)

In this case, A = 7 is distinct from B = 6.

### Step 4: Calculate A + B
Now that we have found A = 7 and B = 6, we can calculate:
\[ A + B = 7 + 6 = 13 \]

### Final Answer
The value of \( A + B \) is **13**.

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