Knights always tell the truth. Rogues always lie. There are 100 people on an island, and each is either a knight or a rogue. Four of them make statements:Alice: "The number of Knights is a multiple of four."Bob: "Exactly one out of Alice, Donald, and I is a Rogue."Carl: "Alice and I are of the same type."Donald: "There are at most 90 Knights."Let X be the number of Knights there are. How many possible values are there for X?a. 2b. 3c. 4d. 5e. 6f. 7g. 8

To solve this problem, we need to analyze each statement one by one.

1. Alice: "The number of Knights is a multiple of four." If Alice is a knight, then the number of knights is a multiple of four. If Alice is a rogue, then the number of knights is not a multiple of four.

2. Bob: "Exactly one out of Alice, Donald, and I is a Rogue." If Bob is a knight, then exactly one of Alice, Donald, and Bob is a rogue. If Bob is a rogue, then more than one or none of Alice, Donald, and Bob is a rogue.

3. Carl: "Alice and I are of the same type." If Carl is a knight, then Alice is also a knight. If Carl is a rogue, then Alice is also a rogue.

4. Donald: "There are at most 90 Knights." If Donald is a knight, then there are at most 90 knights. If Donald is a rogue, then there are more than 90 knights.

Now, let's analyze the statements together. If Alice is a knight, then the number of knights is a multiple of four. If Alice is a rogue, then Carl must also be a rogue. But if Carl is a rogue, then Alice cannot be a rogue. So, Alice must be a knight. Therefore, the number of knights is a multiple of four.

If Bob is a knight, then exactly one of Alice, Donald, and Bob is a rogue. But we know that Alice is a knight, so either Donald or Bob is a rogue. If Donald is a rogue, then there are more than 90 knights. But this contradicts the fact that the number of knights is a multiple of four. So, Donald must be a knight, and Bob must be a rogue.

Therefore, the possible values for X (the number of knights) are multiples of four from 0 to 100, excluding 92, 96, and 100. These are 0, 4, 8, 12, ..., 88, and 90. So, there are 23 possible values for X. The answer is not listed in the options.