The product of two consecutive positive integers is 19 more than their sum. Find the integers.

The crucial part of the wording of the problem tells us that the two integers are consecutive. So, let's call the first term x. Then the other term must be x + 1 since it comes right after it and is an integer. Now we can solve:

x (x + 1) = x + (x + 1) + 19

x^2 + x = 2x + 20

x^2 - x - 20 = 0

Now we factor:

(x - 5) (x + 4) = 0

This means x either equals 5 or x = - 4 to satisfy the above equation. We know that x is positive from the question itself, so x must be 5. The number consecutively after it then must be 6.