4. All natural numbers, the sum of the digits in the representation of which is divisible by 5, are listed in ascending order: $5,14,19,23,28,32, \ldots$ What is the smallest positive difference between consecutive numbers in this sequence? Provide an example and explain why it cannot be smaller.

---

The smallest positive difference between consecutive numbers in this sequence is 1. An example is the pair of numbers 14 and 15. The sum of the digits of 14 is $1 + 4 = 5$, which is divisible by 5. The sum of the digits of 15 is $1 + 5 = 6$, which is not divisible by 5. However, the next number in the sequence is 19, and the sum of its digits is $1 + 9 = 10$, which is divisible by 5. Therefore, the smallest difference is 1, as seen between 14 and 15.

To explain why it cannot be smaller: The smallest possible difference between two numbers is 1. If the difference were 0, the numbers would be the same, which is not possible for consecutive numbers in the sequence. If the difference were greater than 1, there would be a gap between the numbers, which means there would be a number in between that could potentially have a sum of digits divisible by 5, contradicting the requirement that the sequence lists all such numbers in ascending order. Therefore, the smallest possible difference is 1.