Problem 1.
What is the time complexity of this piece of code?
def factorial(n):
r = 1
i = n
while i > 0:
r = r*i
i -= 1
return r

The time complexity of the given code can be analyzed by considering the number of iterations the loop will execute in relation to the input size `n`.

### Code Analysis:
```python
def factorial(n):
    r = 1
    i = n
    while i > 0:
        r = r * i
        i -= 1
    return r
```

1. **Initialization:**
   - `r = 1` and `i = n` are initialized once, which takes constant time, \( O(1) \).

2. **Loop:**
   - The `while` loop executes as long as `i > 0`. Initially, `i = n`, and in each iteration, `i` is decremented by 1 (`i -= 1`).
   - The loop runs from `i = n` down to `i = 1`, meaning it executes `n` times.
   - Each iteration involves a multiplication (`r = r * i`) and a decrement (`i -= 1`), both of which are constant-time operations, \( O(1) \).

3. **Return Statement:**
   - Returning the value of `r` takes constant time, \( O(1) \).

### Time Complexity:
- The loop runs `n` times, and each iteration takes constant time.
- Therefore, the overall time complexity is \( O(n) \).

**Conclusion:**
The time complexity of the `factorial` function is \( O(n) \).

Would you like more details on this analysis or have any questions?

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Here are 5 related questions you might find interesting:

1. How does the time complexity of an iterative factorial algorithm compare to a recursive one?
2. What is the space complexity of the `factorial` function?
3. How would the time complexity change if we added memoization to the factorial function?
4. Can you explain the difference between worst-case and average-case time complexity?
5. How would you modify the factorial function to handle very large numbers?

**Tip:** Always analyze both time and space complexity when assessing an algorithm's efficiency.

Explore the time complexity analysis of the Python factorial function. Learn about algorithm analysis, time complexity, and efficiency. Suitable for advanced high school students and programming enthusiasts.

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