🔢 Euclidean Division vs Floor Division in Python

• Floor Division (//): Rounds the result of division toward negative infinity.
• Euclidean Division: Ensures the remainder is always non-negative, adjusting the quotient if necessary.

🧠 Python Behavior

• Python’s // operator performs floor division.
• Python’s % operator adjusts the remainder to be non-negative when the divisor is positive, making it behave like Euclidean division in many cases.

✅ Example with Positive Divisor

• -7 // 3 → -3
• -7 % 3 → 2
• -7 = 3 × (-3) + 2 → Matches Euclidean division

⚠️ Example with Negative Divisor

• -7 // -3 → 2
• -7 % -3 → -1
• -7 = (-3) × 2 + (-1) → Not Euclidean (remainder is negative)

🧮 Euclidean Quotient Function

def euclidean_quotient(a, b):
    q = a // b
    r = a % b
    return q if r >= 0 else q + 1

✅ Summary Table

💡 Final Insight

• Use // and % together for Euclidean-like behavior when the divisor is positive.
• For consistent Euclidean division regardless of sign, use a custom function like euclidean_quotient(a, b).
• Python’s // performs floor division, which may differ from Euclidean division when negative divisors are involved.
• Understanding the distinction helps avoid subtle bugs in modular arithmetic, cryptography, and number theory applications.