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, let's solve this step by step.

Step 1: We know that the 3-digit number is 4a3 and it is added to 984 to give a 4-digit number 13b7.

Step 2: We can write this as an equation: 4a3 + 984 = 13b7.

Step 3: We also know that the 4-digit number 13b7 is divisible by 11. 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 4: Let's apply this rule to 13b7. The sum of digits at odd places is 1 + b = 1 + b. The sum of digits at even places is 3 + 7 = 10.

Step 5: The difference between these two sums is (1 + b) - 10 = b - 9.

Step 6: For 13b7 to be divisible by 11, b - 9 must be either 0 or a number divisible by 11. The only single digit number that satisfies this condition is 9. So, b = 9.

Step 7: Now, let's substitute b = 9 in the equation from step 2: 4a3 + 984 = 1397.

Step 8: Simplify this equation to find the value of a: 4a3 = 1397 - 984 = 413.

Step 9: The hundreds digit of 413 is 4, which is the same as the hundreds digit of 4a3. So, a = 1.

So, the 3-digit number 4a3 is 413 and the 4-digit number 13b7 is 1397.