🎭 The Poetry of Primes: Divisibility Basics
Once upon a time, numbers learned to share…
🍕 The Big Idea: What is Divisibility?
Imagine you have 12 cookies and want to share them equally with friends. Can you share with 2 friends? Yes! Each gets 6. What about 5 friends? Hmm… that leaves crumbs!
Divisibility is simply asking: “Can this number be shared perfectly, with nothing left over?”
The Magic Rule: A number divides another if it fits in perfectly—no remainders, no leftovers, no crumbs!
Quick Check ✨
- 12 ÷ 3 = 4 ✅ (Perfect! 3 divides 12)
- 12 ÷ 5 = 2 remainder 2 ❌ (Leftovers! 5 does NOT divide 12)
🎁 Factors and Multiples: Best Friends Forever
Think of factors and multiples like a gift-giving relationship:
Factors = The Givers 🎁
Factors are numbers that fit perfectly inside another number.
Example: What are the factors of 12?
- 1 × 12 = 12 ✓
- 2 × 6 = 12 ✓
- 3 × 4 = 12 ✓
Factors of 12: 1, 2, 3, 4, 6, 12
Think of it like this: Which teams can line up evenly to make 12 people?
Multiples = The Receivers 📬
Multiples are numbers you get when you multiply by whole numbers.
Example: Multiples of 4:
- 4 × 1 = 4
- 4 × 2 = 8
- 4 × 3 = 12
- 4 × 4 = 16
- … and so on forever!
Multiples are like the “skip-counting” numbers: 4, 8, 12, 16, 20…
graph TD A[12] --> B[Factors GO INTO 12] A --> C[Multiples COME FROM 12] B --> D[1, 2, 3, 4, 6, 12] C --> E[12, 24, 36, 48...]
➗ The Division Algorithm: The Sharing Formula
When you divide, there’s a beautiful formula that ALWAYS works:
Dividend = Divisor × Quotient + Remainder
Let’s break it down with a story!
🍬 The Candy Story
You have 23 candies and want to share with 5 friends.
- Dividend = 23 (total candies)
- Divisor = 5 (number of friends)
- Quotient = 4 (each friend gets 4)
- Remainder = 3 (leftover candies)
Check: 5 × 4 + 3 = 20 + 3 = 23 ✅
The Golden Rules:
- Remainder is always smaller than divisor
- Remainder is always 0 or positive
- When remainder = 0, we have perfect divisibility!
⭐ Prime Numbers: The Loners
Prime numbers are special numbers with exactly 2 factors: 1 and themselves.
They’re like people who only shake hands with themselves and number 1!
The First 10 Primes:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29
Why is 2 special? 👑
2 is the only even prime! Every other even number can be divided by 2, so they have more than 2 factors.
Is 1 prime? 🤔
NO! 1 only has ONE factor (itself). Primes need EXACTLY two factors.
Quick Prime Test:
To check if a number is prime, try dividing by primes up to its square root!
Is 29 prime?
- √29 ≈ 5.4, so check: 2, 3, 5
- 29 ÷ 2 = 14.5 ❌
- 29 ÷ 3 = 9.67 ❌
- 29 ÷ 5 = 5.8 ❌
- 29 is PRIME! ⭐
🧱 Composite Numbers: The Social Butterflies
Composite numbers have more than 2 factors. They can be broken into smaller pieces!
If primes are loners, composites are the party animals—they have many friends (factors)!
Examples:
| Number | Factors | Type |
|---|---|---|
| 4 | 1, 2, 4 | Composite |
| 6 | 1, 2, 3, 6 | Composite |
| 9 | 1, 3, 9 | Composite |
| 12 | 1, 2, 3, 4, 6, 12 | Composite |
The Number Family:
graph TD A[All Numbers > 1] --> B[Prime Numbers] A --> C[Composite Numbers] B --> D[Exactly 2 factors] C --> E[More than 2 factors] D --> F[2, 3, 5, 7, 11...] E --> G[4, 6, 8, 9, 10...]
Fun Fact: Every composite number can be written as a product of primes! This is called the Fundamental Theorem of Arithmetic.
Example: 12 = 2 × 2 × 3
⚖️ Parity: The Even-Odd Dance
Parity simply means: Is a number even or odd?
Even Numbers 🔵
- Divisible by 2
- End in: 0, 2, 4, 6, 8
- Examples: 2, 4, 6, 8, 10, 100, 1000
Odd Numbers 🔴
- NOT divisible by 2
- End in: 1, 3, 5, 7, 9
- Examples: 1, 3, 5, 7, 9, 11, 101
Parity Magic Tricks ✨
| Operation | Result |
|---|---|
| Even + Even | Even (2+4=6) |
| Odd + Odd | Even (3+5=8) |
| Even + Odd | Odd (2+3=5) |
| Even × Any | Even |
| Odd × Odd | Odd |
🔮 Divisibility Rules: The Shortcut Spells
Why divide when you can use magic? These rules let you check divisibility at a glance!
÷2: The Last Digit Spell
A number is divisible by 2 if its last digit is even (0, 2, 4, 6, 8)
✅ 1,234 → ends in 4 → divisible by 2 ❌ 1,237 → ends in 7 → NOT divisible by 2
÷3: The Digit Sum Spell
A number is divisible by 3 if the sum of its digits is divisible by 3
✅ 123 → 1+2+3 = 6 → 6÷3=2 ✓ ❌ 124 → 1+2+4 = 7 → 7÷3=2.33 ✗
÷4: The Last Two Digits Spell
A number is divisible by 4 if its last 2 digits form a number divisible by 4
✅ 1,324 → 24÷4=6 ✓ ❌ 1,326 → 26÷4=6.5 ✗
÷5: The Ending Spell
A number is divisible by 5 if it ends in 0 or 5
✅ 125, 130, 500 ❌ 123, 127
÷6: The Double Check
A number is divisible by 6 if it’s divisible by BOTH 2 and 3
✅ 126 → even AND 1+2+6=9 (÷3) ✓ ❌ 124 → even BUT 1+2+4=7 (not ÷3) ✗
÷9: The Super Digit Sum
A number is divisible by 9 if the sum of digits is divisible by 9
✅ 729 → 7+2+9 = 18 → 1+8 = 9 ✓ ❌ 730 → 7+3+0 = 10 ✗
÷10: The Zero Ending
A number is divisible by 10 if it ends in 0
✅ 120, 1000, 50 ❌ 125, 101
🔢 Digit Sums: The Number’s DNA
The digit sum is what you get when you add all the digits of a number together.
How to Calculate:
Example: 5,847
- Step 1: 5 + 8 + 4 + 7 = 24
- Step 2 (optional): 2 + 4 = 6 (this is the “digital root”)
Why Digit Sums Matter:
- Divisibility by 3: If digit sum ÷ 3 = whole number ✓
- Divisibility by 9: If digit sum ÷ 9 = whole number ✓
- Quick mental math: Helps verify calculations!
The Digital Root Trick 🎩
Keep adding until you get a single digit:
- 999 → 9+9+9 = 27 → 2+7 = 9
- 1234 → 1+2+3+4 = 10 → 1+0 = 1
- 8888 → 8+8+8+8 = 32 → 3+2 = 5
🎯 Putting It All Together
Let’s analyze the number 360:
| Property | Check | Result |
|---|---|---|
| Divisible by 2? | Ends in 0 | ✅ Yes |
| Divisible by 3? | 3+6+0=9 | ✅ Yes |
| Divisible by 4? | 60÷4=15 | ✅ Yes |
| Divisible by 5? | Ends in 0 | ✅ Yes |
| Divisible by 6? | ÷2 and ÷3 | ✅ Yes |
| Divisible by 9? | Digit sum=9 | ✅ Yes |
| Divisible by 10? | Ends in 0 | ✅ Yes |
| Prime or Composite? | Many factors | Composite |
| Parity? | Even | Even |
| Digit Sum | 3+6+0 | 9 |
360 is a very “friendly” number—it divides by almost everything!
🌟 Key Takeaways
graph LR A[Divisibility Basics] --> B[Divisibility] A --> C[Factors & Multiples] A --> D[Division Algorithm] A --> E[Prime Numbers] A --> F[Composite Numbers] A --> G[Parity] A --> H[Divisibility Rules] A --> I[Digit Sums] B --> B1[No remainder = divides] C --> C1[Factors go IN<br>Multiples come OUT] D --> D1[Dividend = Divisor × Quotient + Remainder] E --> E1[Exactly 2 factors] F --> F1[More than 2 factors] G --> G1[Even or Odd] H --> H1[Quick checks without dividing] I --> I1[Add all digits together]
“In the world of numbers, divisibility is the art of perfect sharing. Master it, and you hold the key to understanding how all numbers are connected!”
🎉 Congratulations! You now have the foundation to explore the beautiful world of Number Theory!