A three digit number 4a3 is added to another three digit number 984 to give a four digit number 13b7, which is divisible by 11, then find the value of a.

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 multiple of 11.

Step 2: In the number 13b7, the sum of digits at odd places = 1 + b and the sum of digits at even places = 3 + 7 = 10.

Step 3: So, 1 + b - 10 = b - 9 should be a multiple of 11.

Step 4: The possible values of b that make b - 9 a multiple of 11 are 0, 11, 22, 33, ..., but since b is a digit, its value must be between 0 and 9. So, the only possible value for b is 0.

Step 5: Now, we know that 4a3 + 984 = 13b7. Substituting b = 0, we get 4a3 + 984 = 1307.

Step 6: Subtracting 984 from both sides, we get 4a3 = 323.

Step 7: Comparing the hundreds and tens places, we see that a = 2.

So, the value of a is 2.