The difference of two natural numbers is 3, and their product is 87 more than their sum. Find these numbers.

Let us denote the sought numbers by the letters x and y, then the condition of the problem can be written in the form of the following system of equations:

x – y = 3,

x * y – 87 = x + y.

From the first equation we get that y = x – 3. Substitute the resulting value of y into the second equation and get:

x * (x – 3) – 87 = x + x – 3,

x² – 3 * x – 87 = 2 * x – 3,

x² – 5 * x – 84 = 0.

The discriminant of this quadratic equation will be:

(-5) ² – 4 * 1 * (-84) = 361.

Therefore, the equation has the following roots:

x = (5 – 19) / 2 = -7 and x = (5 + 19) / 2 = 12.

Since, according to the condition of the problem, the numbers must be natural, we have a unique solution:

x = 12 y = 12 – 3 = 9.