answer:1. **Initial Setup and Recursive Formula**: Let ( y ) be the initial number of coins. The sequence of shares and remaining coins can be expressed as follows: - First pirate takes frac{1}{15}y, leaving y_1 = frac{14}{15}y. - The pattern continues such that after the ( k^{text{th}} ) pirate, the remaining coins are: [ y_k = frac{15-k}{15}y_{k-1} ] 2. **Expression for the 15^text{th} Pirate**: The 15^text{th} pirate takes all remaining coins, y_{14}. Compute y_{14} using: [ y_{14} = left(frac{1}{15}right) left(frac{2}{15}right) ldots left(frac{14}{15}right) y = frac{14!}{15^{14}} y ] 3. **Finding the Smallest y**: For y_{14} to be an integer, ( y ) must be a multiple of the denominator when frac{14!}{15^{14}} is reduced: [ y = k cdot 15^{14} / GCD(14!, 15^{14}) quad text{for some integer } k ] Calculating the factors: [ 15^{14} = (3 cdot 5)^{14} = 3^{14} cdot 5^{14} ] [ 14! text{ factors into } 3^5 cdot 5^2 text{ (along with other primes)} ] So the least ( y ) such that all shares are integers is: [ y = 15^{14} / (3^{5} cdot 5^{2}) ] 4. **Conclusion**: The smallest adequate ( y ) gives us the least number of coins initially, thus the 15^text{th} pirate receives: [ 15^{7} ] because the simplified expression becomes y = 3^9 cdot 5^{12}, so the last pirate takes y_{14} = 15^7. The final answer is boxed{textbf{(A)} 15^7}

answer:To determine which sentence is nonsensical, let's analyze each sentence individually. Sentence A: He went to prison for murder This sentence makes sense because people can go to prison as a consequence of committing a crime such as murder. Sentence B: He went to prison for charity This sentence is nonsensical because going to prison is generally not associated with charitable acts. People typically go to prison as a consequence of committing a crime, not for participating in charity. Therefore, the nonsensical sentence is Sentence B: He went to prison for charity.

answer:The candidate's popularity led to a wide victory.

answer:This is due to how Python's iteration protocol works. When you iterate over a list in the forward direction, Python internally keeps track of the current index. If you remove an element while iterating, the indices of the remaining elements shift down by one, but the iterator doesn't know about this change. As a result, it will skip an element or visit an index that no longer exists, causing a `RuntimeError`. However, when you iterate over a list in reverse order, the situation is different. When you remove an element, the indices of the elements before it don't change, so the iterator remains valid. Here is an example: ```python # Forward iteration my_list = [1, 2, 3, 4, 5] for i in my_list: my_list.remove(i) # This will raise a RuntimeError # Reverse iteration my_list = [1, 2, 3, 4, 5] for i in reversed(my_list): my_list.remove(i) # This will not raise an exception ``` Note that while iterating in reverse order works in this case, it's still generally not recommended to modify a list while iterating over it. It's usually safer and more readable to create a copy of the list or use a list comprehension to avoid modifying the original list. ```python # Creating a copy my_list = [1, 2, 3, 4, 5] for i in my_list[:]: my_list.remove(i) # List comprehension my_list = [1, 2, 3, 4, 5] my_list = [i for i in my_list if i not in my_list] ```