7.4. On a certain island, there live 100 people, each of whom is either a knight, who always tells the truth, or a liar, who always lies. One day, all the inhabitants of this island lined up, and the first one said: "The number of knights on this island is a divisor of the number 1." Then the second said: "The number of knights on this island is a divisor of the number 2," and so on until the hundredth, who said: "The number of knights on this island is a divisor of the number 100." Determine how many knights can live on this island. (Find all answers and prove that there are no others.)
Solution: If there are no knights, then all the speakers are lying, since 0 is not a divisor of any natural number.

If there are knights, let there be $a$ of them. Then only the people with numbers $a k$ for $k=1,2, \ldots$ are telling the truth. On the other hand, since exactly $a$ people are telling the truth, $k$ changes precisely from 1 to $a$.

Thus, $a$ is such a number that the person with number $a \cdot a$ is still in the row $\left(a^{2} \leqslant 100\right)$, while the person with number $a \cdot(a+1)$ is not $(a(a+1)>100)$. Clearly, only $a=10$ fits, as for $a10$ - the first.

In total, there are either 0 or 10 knights.

Criteria: Only the answer - 0 points.

The case where there are no knights - 1 point.

Noted that the people with numbers $a, 2 a, \ldots, k a$ should be telling the truth - 2 points.

Proven that $k=a$ - another 2 points.

Proven that then $a=10$ - another 2 points.