A sandwich and a meal plate cost $\mathrm{R} \$ 5.00$ and $\mathrm{R} \$ 7.00$, respectively. In how many ways can one buy only sandwiches, only meal plates, or some combination of sandwiches and meal plates with $\mathrm{R} \$ 90.00$, without leaving any change?
If $S$ corresponds to the number of sandwiches and $P$ to the number of meal plates, then $5 S+7 P=90$ and, therefore,

$$
P=\frac{90-5 S}{7}=5 \times \frac{18-S}{7}
$$

Since we want non-negative integer solutions for $P$ and $S$, we see that 7 must divide $18-S$. Thus, $S$ can only be 4, 11, or 18, and in these cases, $P$ is equal to 10, 5, or 0, respectively. Therefore, we have only three ways to make the purchase without receiving change, namely, 4 sandwiches and 10 plates, 11 sandwiches and 5 plates, or 18 sandwiches and no plates.