5. How many solutions in natural numbers $x, y$ does the inequality $x / 76 + y / 71 < 1$ have?

(12 points)
# Solution.

All solutions lie in the rectangle

$$
\Pi=\{0<x<76 ; \quad 0<y<41\}
$$

We are interested in the number of integer points lying inside Π below its diagonal $x / 76 + y / 41=1$. There are no integer points on the diagonal itself, since 76 and 41 are coprime. So we get half the number of integer points inside Π:

$$
\frac{75 \cdot 40}{2}=1500
$$

Answer: 1500