## Task B-2.4.

The teacher has several candies that she wants to distribute to students who come for additional math lessons so that each student gets an equal number of candies and no candy is left over.

- If two students come, each student cannot get the same number of candies.
- If three students come, each of them will get the same number of candies.
- If four students come, each of them will get the same number of candies.
- If five students come, each of them will get the same number of candies.
- If six students come, each student cannot get the same number of candies.

If it is known that only one of the above statements is false, what is the minimum number of candies the teacher can have?
## Solution.

Let $x$ be the sought number of candies. Then the given statements can be written in the following form:

1. $x$ is not a multiple of 2;
2. $x$ is a multiple of 3;
3. $x$ is a multiple of 4;
4. $x$ is a multiple of 5;
5. $x$ is not a multiple of 6.

If the first statement is not true, then all statements from 2. to 5. are true, which means that $x$ is a multiple of 2 and 3, which is in contradiction with the fact that it is not a multiple of 6. This means that the first statement is true.

Since the first statement is true, $x$ is an odd number, so the third statement must be false, and all the others must be true.

We conclude that $x$ is an odd multiple of 3 and 5.

The smallest number of candies the teacher can have is 15, as we are looking for the smallest odd common multiple of 3 and 5.

Note: A student can also come to the conclusion that the third statement is false through some other logical reasoning. Correct reasoning of this fact is worth 4 points.