There are 23 two-digit numbers that are divisible by the sum of their digits.
Let's examine how to find these numbers. A two-digit number can be represented as 10a + b, where 'a' is the tens digit and 'b' is the units digit, and both 'a' and 'b' are integers such that 1 ≤ a ≤ 9 and 0 ≤ b ≤ 9. We need to find the numbers for which (10a + b) is divisible by (a + b). This means that (10a + b) / (a + b) results in an integer.
We can systematically check each two-digit number:
- 10: 10 / (1+0) = 10 / 1 = 10. Divisible.
- 11: 11 / (1+1) = 11 / 2 = 5.5. Not divisible.
- 12: 12 / (1+2) = 12 / 3 = 4. Divisible.
- 13: 13 / (1+3) = 13 / 4 = 3.25. Not divisible.
- 14: 14 / (1+4) = 14 / 5 = 2.8. Not divisible.
- 15: 15 / (1+5) = 15 / 6 = 2.5. Not divisible.
- 16: 16 / (1+6) = 16 / 7 ≈ 2.29. Not divisible.
- 17: 17 / (1+7) = 17 / 8 ≈ 2.13. Not divisible.
- 18: 18 / (1+8) = 18 / 9 = 2. Divisible.
- 19: 19 / (1+9) = 19 / 10 = 1.9. Not divisible.
Continuing this process for all two-digit numbers from 10 to 99, we find the following numbers satisfy the condition:
10, 12, 18, 20, 21, 24, 27, 30, 36, 40, 42, 45, 48, 50, 54, 60, 63, 70, 72, 80, 81, 84, 90
Counting these numbers, we find there are 23 such numbers.
