he-024

# HumanEval/24

Implement a file **`solution.py`** that completes the function below. Keep the given name and signature; your file must define it at module level.

```python


def largest_divisor(n: int) -> int:
    """ For a given number n, find the largest number that divides n evenly, smaller than n
    >>> largest_divisor(15)
    5
    """
```

<!-- imported from OpenAI HumanEval (HumanEval/24) -->

# Auto-generated from OpenAI HumanEval HumanEval/24. Do not edit by hand.
from solution import *  # noqa: F401,F403 (prompt helpers may be referenced by tests)
from solution import largest_divisor as candidate

METADATA = {
    'author': 'jt',
    'dataset': 'test'
}


def check(candidate):
    assert candidate(3) == 1
    assert candidate(7) == 1
    assert candidate(10) == 5
    assert candidate(100) == 50
    assert candidate(49) == 7


def test_he_024():
    check(candidate)

You are an expert programmer. Solve the task exactly as specified. Output your solution as fenced code blocks using the required file name(s) and the exact function/type signatures requested. Prefer correctness; do not include prose outside code unless asked.

Looking at this problem, I need to find the largest divisor of `n` that is smaller than `n` itself.

Let me think through the approach:
1. For any number n > 1, I want to find the largest proper divisor
2. The largest proper divisor would be n divided by its smallest prime factor
3. For example, with n=15: factors are 1,3,5,15. The largest proper divisor is 5 (which is 15/3)
4. For a prime number like 7: the only proper divisor is 1

I can optimize this by checking divisors from sqrt(n) downwards to find the largest one efficiently.

Actually, let me think more carefully:
- I want the largest divisor d where d < n and n % d == 0
- This is equivalent to finding the smallest prime factor of n, then dividing n by that factor

Let me trace through examples:
- n=15: factors are 1,3,5,15. Largest proper divisor = 5 (15/3)
- n=12: factors are 1,2,3,4,6,12. Largest proper divisor = 6 (12/2)  
- n=7: prime, factors are 1,7. Largest proper divisor = 1

Here's my approach:
1. Check for divisors from 2 up to sqrt(n)
2. When I find the first divisor d, then n/d is the largest proper divisor
3. If no divisor found, n is prime and answer is 1

```python
def largest_divisor(n: int) -> int:
    """ For a given number n, find the largest number that divides n evenly, smaller than n
    >>> largest_divisor(15)
    5
    """
    if n <= 1:
        return None
    
    # Find the smallest divisor >= 2
    for i in range(2, int(n**0.5) + 1):
        if n % i == 0:
            return n // i
    
    # If no divisor found, n is prime, so largest proper divisor is 1
    return 1
```