A three digit number 4a3 is added to another three digit number 984 to give a…

Divisibility rule for 11: a number is divisible by 11 exactly when the alternating sum of its digits, taken from the right, is 0 or a multiple of 11. Column addition fixes every digit of a sum uniquely once the carries into each column are decided, so those carries can be checked against the digits actually given in the sum.

1. Add the units column: 3 + 4 = 7, which matches the units digit of 13b7 with no carry into the tens column.

2. Add the hundreds column: 4 + 9 = 13. Since the thousands digit of the sum is 1 and the hundreds digit is 3, this accounts for the "13" exactly, which means no extra carry arrived from the tens column — i.e., the tens column addition a + 8 does not reach 10.

3. So the tens column gives a + 8 = b directly, with no carry, linking the two unknown digits.

4. Apply the divisibility-by-11 rule to 13b7: taking digits from the right, (7 + 3) - (b + 1) must be 0 or a multiple of 11, i.e., 9 - b = 0 (the only value that keeps b a single digit), so b = 9.

5. Substitute b = 9 into a + 8 = b to get a = 1.

Cross-check: direct addition confirms it — 413 + 984 = 1397, and 1397 ÷ 11 = 127 exactly, so the digits a = 1 and b = 9 are consistent with both the column addition and the divisibility condition.

Therefore a + b = 1 + 9 = 10.