A group of 45 tokens is arranged into three piles such that each pile contains a different prime number of tokens. What is the greatest number of tokens possible in any of the three piles?
To find the greatest number of tokens possible in any of the three piles, we need to consider the largest prime numbers smaller than or equal to 45.
The prime numbers smaller than or equal to 45 are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, and 43.
Since each pile must have a different prime number of tokens, we can start with the largest prime number, which is 43. For the second pile, we can choose the next largest prime number, which is 41.
Now, we have 43 + 41 = 84 tokens distributed in two piles. To find the number of tokens left for the third pile, we subtract this sum from the total number of tokens: 45 - 84 = -39.
Since we cannot have a negative number of tokens, we need to adjust our distribution. Let's try a different approach.
We can allocate the tokens as follows:
- The first pile has 43 tokens (the largest prime number).
- The second pile has 5 tokens (a different prime number).
- The third pile has 45 - (43 + 5) = -3 tokens.
Again, we have a negative number, so we adjust the distribution.
Finally, we can allocate the tokens as follows:
- The first pile has 41 tokens (the largest prime number less than 45).
- The second pile has 3 tokens (a different prime number).
- The third pile has 45 - (41 + 3) = 1 token.
Therefore, the greatest number of tokens possible in any of the three piles is 41.