When a 3 digit number 984 is added to another 3 digit number 4 p3, we get a four digit number 13q7, which is divisible by 11. The value of `p + q` is :

Options

A. 10
B. 11
C. 12
D. 13

Answer

To solve the problem step by step, we need to follow the given information carefully.

### Step 1: Understand the numbers involved
We have two numbers:
- The first number is 984.
- The second number is represented as 4p3, where p is a digit we need to find.

When we add these two numbers, we get a four-digit number represented as 13q7, where q is another digit we need to find.

### Step 2: Set up the equation
We can express the addition as:
\[ 984 + 4p3 = 13q7 \]

### Step 3: Calculate the sum of the numbers
First, we need to convert 4p3 into a numerical form:
- 4p3 = 400 + 10p + 3

Now, we can rewrite the equation:
\[ 984 + (400 + 10p + 3) = 1300 + 10q + 7 \]

This simplifies to:
\[ 984 + 403 + 10p = 1300 + 10q + 7 \]
\[ 1387 + 10p = 1300 + 10q \]

### Step 4: Rearranging the equation
Now, we can rearrange the equation to isolate terms involving p and q:
\[ 10p - 10q = 1300 - 1387 \]
\[ 10p - 10q = -87 \]

Dividing the entire equation by 10 gives:
\[ p - q = -8.7 \]

Since p and q are digits, we can express this as:
\[ p - q = -8 \]
or
\[ p = q - 8 \]

### Step 5: Use the divisibility rule of 11
Next, we need to check the divisibility of the number 13q7 by 11. According to the rule of divisibility for 11, the difference between the sum of the digits in odd positions and the sum of the digits in even positions should be either 0 or a multiple of 11.

The digits in 13q7 are:
- Odd positions: 1 (1st) + q (3rd) + 7 (4th) = 1 + q + 7 = q + 8
- Even positions: 3 (2nd) + 7 (4th) = 3 + 7 = 10

Now, we calculate the difference:
\[ (q + 8) - 10 = q - 2 \]

### Step 6: Set up the divisibility condition
For the number to be divisible by 11:
\[ q - 2 = 0 \quad \text{or} \quad q - 2 = 11k \quad (k \text{ is an integer}) \]

The only feasible value for q (since it is a digit) is:
\[ q - 2 = 0 \]
Thus,
\[ q = 2 \]

### Step 7: Substitute q back to find p
Now that we have q, we can find p using the earlier equation:
\[ p = q - 8 \]
\[ p = 2 - 8 = -6 \]
This is not a valid digit. Therefore, we need to check for other values of q that satisfy the divisibility condition.

### Step 8: Check for valid values of q
Let's check for other values of q:
If we set \( q = 9 \):
\[ p = 9 - 8 = 1 \]

### Step 9: Calculate p + q
Now we have:
- p = 1
- q = 9

Thus, the value of \( p + q \) is:
\[ p + q = 1 + 9 = 10 \]

### Final Answer
The value of \( p + q \) is **10**.

---