Problem 3. There were several whole cheese heads on the warehouse. At night, rats came and ate 10 heads, and everyone ate equally. Several rats got stomachaches from overeating. The remaining 7 rats the next night finished off the remaining cheese, but each rat could eat only half as much cheese as the night before. How much cheese was originally on the warehouse? [5 points] (A.D. Blyunkov, I. V. Raskina)
Answer: 11 cheese heads

Solution: Let the total number of rats be $k$ ( $k>7$ ), then each rat ate $10 / k$ cheese heads on the first night. On the second night, each rat ate half as much, that is, $5 / k$ cheese heads. Thus, the 7 rats ate a total of $35 / k$ cheese heads. This is an integer. The only divisor of the number 35 that exceeds $7$ is the number 35 itself. Therefore, $35 / k=1$, and the total number of cheese heads on the warehouse before the rat invasion was $10+1=11$ cheese heads.