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.

Sure, I understand. Let's solve this step by step.

Step 1: We know that a number is divisible by 11 if the difference between the sum of its digits at odd places and the sum of its digits at even places is either 0 or a number divisible by 11.

Step 2: The 4-digit number is 13b7. Let's apply the rule from step 1. The sum of digits at odd places = 1 + b and the sum of digits at even places = 3 + 7 = 10.

Step 3: The difference between the sum of digits at odd and even places is |(1 + b) - 10|. This must be divisible by 11.

Step 4: Simplifying gives us |b - 9|. The absolute value of a number is always positive, and the only single digit number that when subtracted from 9 gives a result divisible by 11 is 0. So, b = 9.

Step 5: Now, let's find the value of a. We know that 4a3 + 984 = 13b7. Substituting b = 9 gives us 4a3 + 984 = 1397.

Step 6: Subtracting 984 from both sides gives us 4a3 = 413. So, a = 1.

So, the values of a and b are 1 and 9 respectively.