To solve this problem, you can use the fact that the sum of the digits of a number is equal to the remainder when that number is divided by 9. This is because the digits of a number represent multiples of powers of 10, and the sum of these multiples will be equal to the remainder when the number is divided by 9.
For example, consider the number 12345. The sum of the digits is 1 + 2 + 3 + 4 + 5 = 15. Dividing 12345 by 9 gives a remainder of 6, which is the same as the sum of the digits.
Using this fact, you can convert the problem of finding the number of positive integers less than 1,000,000 with a sum of digits equal to 19 into a problem of finding the number of positive integers less than 1,000,000 with a remainder of 19 when divided by 9.
To do this, you can start by finding the remainder when 1,000,000 is divided by 9. Since 1,000,000 is divisible by 9, the remainder will be 0. Therefore, the remainder of any positive integer less than 1,000,000 when divided by 9 will be an integer between 0 and 8.
Since 19 is not between 0 and 8, there are no positive integers less than 1,000,000 with a remainder of 19 when divided by 9. Therefore, there are no positive integers less than 1,000,000 with a sum of digits equal to 19.