A 3-digit number 4a3 is added to another 3-digit number 984 to give the four-digit number 13b7 which is divisible by 11 Then (a + b) is A.10B.11C.12D.13

To determine the value of (a + b), we need to analyze the divisibility rules for 11 and use the given information.

Let's break down the information given:

A 3-digit number 4a3 is added to another 3-digit number 984 to give the four-digit number 13b7.
The resulting four-digit number 13b7 is divisible by 11.
Using the divisibility rule for 11, the difference between the sum of the digits in the odd positions and the sum of the digits in the even positions should be divisible by 11.

Let's apply this rule to the four-digit number 13b7:

Sum of the digits in odd positions = 1 + b = b + 1
Sum of the digits in even positions = 3 + 7 = 10

According to the divisibility rule for 11, the difference between these two sums must be divisible by 11. Therefore, we have:

(b + 1) - 10 = b + 1 - 10 = b - 9

Since the resulting four-digit number 13b7 is divisible by 11, the difference (b - 9) must also be divisible by 11. This means that b - 9 can be any multiple of 11, such as 0, 11, 22, 33, and so on.

To find the value of b, we need to check which multiple of 11 will give us a valid 3-digit number when 9 is added to it. We'll go through the possibilities:

If b - 9 = 0, then b = 9. However, 13b7 will not be a valid four-digit number, as b = 9 would result in 1397, which is a three-digit number.

If b - 9 = 11, then b = 20. However, 13b7 will not be a valid four-digit number, as b = 20 would result in 1327, which is a three-digit number.

If b - 9 = 22, then b = 31. However, 13b7 will not be a valid four-digit number, as b = 31 would result in 1337, which is a three-digit number.

If b - 9 = 33, then b = 42. In this case, 13b7 will be a valid four-digit number, as b = 42 results in 1347.

Since b = 42 satisfies the conditions and gives us a valid four-digit number, we can conclude that b = 42.

Now that we know b = 42, we can find the value of a. Looking at the given information, we know that the 3-digit number 4a3 is added to 984 to give the four-digit number 13b7. Therefore:

4a3 + 984 = 13b7

Substituting the value of b, we get:

4a3 + 984 = 1347

Now we can solve for a:

4a3 = 1347 - 984
4a3 = 363

Since a is a single digit, the only value that satisfies this equation is a = 6.

Therefore, (a + b) = 6 + 42 = 48.

So, the correct option is D. 48.