updated the explaination for the solution

Author: Vitessse

Math/#2894 - DIvisible and non-divisible sums difference - Easy/Explaination.md

@@ -8,36 +8,75 @@
 
 ## 📝 Introduction
 
-*A brief introduction to the problem. Mention the key aspects, constraints, and expected output.*
+You're given two integers, n and m. Your task is to calculate the difference between:
+
+num1: the sum of all numbers from 1 to n not divisible by m.
+
+num2: the sum of all numbers from 1 to n divisible by m.
+
+Return the value of num1 - num2.
 
 ---
 
 ## 💡 Approach & Key Insights
 
-*Describe the main idea behind solving the problem. What observations lead to a solution? Mention brute force or naive approaches first, then discuss optimal solutions.*
+The total sum of numbers from 1 to n is known via the formula: n(n + 1) / 2.
+
+The sum of numbers divisible by m is also a simple arithmetic series.
+
+Subtracting 2 * (sum of numbers divisible by m) from the total sum gives us num1 - num2.
+
+
 
 ---
 
 ## 🛠️ Breakdown of Approaches
 
 ### 1️⃣ Brute Force / Naive Approach
 
-- **Explanation:** *Describe the brute force solution and why it works.*
-- **Time Complexity:** *O(?) - Explanation*
-- **Space Complexity:** *O(?) - Explanation*
+-**Explanation:** Loop through all numbers from 1 to n. For each number:
+
+If it is divisible by m, add to num2.
+
+Else, add to num1.
+Finally return num1 - num2.
+- **Time Complexity:** *O(?) - because we loop from 1 to n.*
+- **Space Complexity:** *O(?) - constant space for sums.*
 - **Example/Dry Run:**
 
-Example input: [Insert example] Step 1 → Step 2 → Step 3 → Output
+Example input: n = 10, m = 3
+
+Loop from 1 to 10:
+
+Not divisible by 3: 1 + 2 + 4 + 5 + 7 + 8 + 10 = 37
+
+Divisible by 3: 3 + 6 + 9 = 18
+
+Output: 37 - 18 = 19
 
 
 ### 2️⃣ Optimized Approach
 
-- **Explanation:** *Describe an optimized approach with clear reasoning.*
-- **Time Complexity:** *O(?) - Explanation*
-- **Space Complexity:** *O(?) - Explanation*
+- **Explanation:** Use math formulas to compute the total sum of numbers from 1 to n, and separately the sum of numbers divisible by m:
+
+Total sum: n(n + 1) / 2
+
+Sum divisible by m: m * k(k + 1) / 2 where k = n // m
+
+Return: total_sum - 2 * divisible_sum
+- **Time Complexity:** *O(1) - no loops, only arithmetic.*
+- **Space Complexity:** *O(1) 
 - **Example/Dry Run:**
 
-Example input: [Insert example] Step 1 → Step 2 → Step 3 → Output
+Input: n = 10, m = 3
+
+total_sum = 10 × 11 / 2 = 55
+
+k = 10 // 3 = 3
+
+divisible_sum = 3 × (3 × 4 / 2) = 3 × 6 = 18
+
+Output = 55 - 2 × 18 = 19
 
 
 ### 3️⃣ Best / Final Optimized Approach (if applicable)
@@ -55,41 +94,40 @@ Example input: [Insert example] Step 1 → Step 2 → Step 3 → Output
 
 | Approach      | Time Complexity | Space Complexity |
 | ------------- | --------------- | ---------------- |
-| Brute Force   | O(?)            | O(?)             |
-| Optimized     | O(?)            | O(?)             |
+| Brute Force   | O(n)            | O(1)             |
+| Optimized     | O(1)            | O(1)             |
 | Best Approach | O(?)            | O(?)             |
 
 ---
 
 ## 📉 Optimization Ideas
 
-*Are there any ways to further improve the solution? Can we reduce memory usage or optimize certain operations?*
+The optimized solution already achieves constant time and space using mathematical formulas. No further optimization is necessary unless extending to very large ranges, in which case care for integer overflow may be required.
 
 ---
 
 ## 📌 Example Walkthroughs & Dry Runs
 
-*Use visuals, ASCII diagrams, or step-by-step breakdowns for clarity.*
-
-```plaintext
 Example:
-Input: 1 -> 2 -> 3 -> 4
-Process:
-1 -> Swap 2 & 3
-2 -> Reverse half of list
-3 -> Merge both halves
-Output: 1 -> 3 -> 2 -> 4
+Input: n = 5, m = 2
+Total sum = 5 * 6 / 2 = 15
+Divisible by 2: 2 + 4 = 6
+Not divisible: 1 + 3 + 5 = 9
+Answer: 9 - 6 = 3
+
+Or:
+Answer = 15 - 2 * 6 = 3
 ```
 
 ---
 
 ## 🔗 Additional Resources
 
-- [Resource 1]()
-- [Resource 2]()
-- [Resource 3]()
+- Arithmetic Series - Wikipedia
+- Python Integer Arithmetic
+
 
 ---
 
-Author: Your Name
-Date: DD/MM/YYYY
+Author: Andrew
+Date: 07/06/2025