A 3-digit number 4a3 is added to another 3-digit number 984 to give a 4-digit number 13b7, which is divisible by 11. Then, (a + b) =?

To solve this problem, we need to find the values of 'a' and 'b' that satisfy the given conditions.

First, let's break down the information we have:

We have a 3-digit number, 4a3, and another 3-digit number, 984. When we add these two numbers together, we get a 4-digit number, 13b7, which is divisible by 11.

To determine the value of 'a', we need to consider the divisibility rule for 11. According to this rule, 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.

In the 4-digit number 13b7, the sum of the digits in the odd positions is (1 + b) and the sum of the digits in the even positions is (3 + 7). Since the number is divisible by 11, the difference between these two sums should also be divisible by 11.

Therefore, we can write the equation: (1 + b) - (3 + 7) = 11k, where 'k' is an integer.

Simplifying this equation, we get: 1 + b - 3 - 7 = 11k Simplifying further, we have: b - 9 = 11k

Now, let's consider the value of 'b'. Since 'b' is a digit, it can only take values from 0 to 9. We need to find the value of 'b' that satisfies the equation b - 9 = 11k.

By trying different values of 'b', we can find that when 'b' is equal to 9, the equation becomes: 9 - 9 = 11k, which simplifies to 0 = 11k. This means that 'k' can be any integer.

Therefore, when 'b' is equal to 9, the equation is satisfied.

Now, let's find the value of 'a'. We know that the 4-digit number is 13b7, so 'a' must be the digit between 4 and 3. Since 'b' is equal to 9, 'a' must be 4.

Finally, we need to find the value of (a + b). Substituting the values we found, we have: (4 + 9) = 13.

Therefore, (a + b) is equal to 13.