2. The old man has a certain number of apples and has said the following about them.

- If two of my grandchildren come, I will not be able to divide the apples equally between them.
- If three of my grandchildren come, I will be able to divide the apples equally among all of them.
- If four of my grandchildren come, I will be able to divide the apples equally among all of them.
- If five of my grandchildren come, I will be able to divide the apples equally among all of them.
- If six of my grandchildren come, I will not be able to divide the apples equally among all of them.

However, the old man is a bit forgetful, so exactly one of these statements is not true. Under these conditions, what is the minimum number of apples the old man can have?
2. Let's assume that the old man's first statement is false (and then all the others must be true). This means that the old man has an even number

![](https://cdn.mathpix.com/cropped/2024_06_04_f254ba4f25ce7f8d9df1g-31.jpg?height=404&width=464&top_left_y=1055&top_left_x=1051)

Op 2017 2B 1 of apples, and from the second statement, it follows that the number of apples he has is divisible by 3. Therefore, in this case, the number of apples the old man has is divisible by 6, but then his last statement is not true, a contradiction. Therefore, the old man's first statement must be true, which means that the old man has an odd number of apples. In this case, the third statement is clearly not true, so all the others must be true, i.e., the number of apples must be odd, divisible by 3, divisible by 5, and not divisible by 6. The smallest natural number that is divisible by 3 and 5 is 15, and it clearly meets all the necessary conditions, so the old man can have a minimum of 15 apples.