Introduction to Divisibility Rules
Divisibility rules are shortcuts that help you determine whether one number is divisible by another without performing the actual division. These rules are essential for mental math, simplifying fractions, factoring numbers, and solving various mathematical problems efficiently.
Why Divisibility Rules Matter:
- Speed up mental calculations and estimations
- Simplify fractions by identifying common factors
- Help with prime factorization and number theory
- Essential for competitive exams and mathematical reasoning
- Foundation for more advanced mathematical concepts
In this comprehensive guide, we'll explore divisibility rules for numbers 1 through 13, with clear explanations, practical examples, and interactive tools to help you master these essential mathematical shortcuts.
What is Divisibility?
A number a is divisible by another number b if when you divide a by b, the result is an integer with no remainder. In other words, b divides evenly into a.
This means that a can be expressed as the product of b and some integer c:
Examples:
15 is divisible by 3 because 15 ÷ 3 = 5 (an integer)
28 is divisible by 7 because 28 ÷ 7 = 4 (an integer)
17 is NOT divisible by 5 because 17 ÷ 5 = 3.4 (not an integer)
- Dividend: The number being divided (a)
- Divisor: The number dividing the dividend (b)
- Quotient: The result of the division (c)
- Remainder: What's left over when division isn't exact
- Factor: A divisor that divides a number exactly
Strengthen your problem-solving ability using the divisibility-calculator.
Basic Divisibility Rules (1-5)
These are the most fundamental divisibility rules that everyone should know:
Divisibility by 1
Rule: Every integer is divisible by 1.
Why it works: Any number divided by 1 equals itself.
Examples:
7 ÷ 1 = 7 Divisible
256 ÷ 1 = 256 Divisible
-15 ÷ 1 = -15 Divisible
Divisibility by 2
Rule: A number is divisible by 2 if its last digit is even (0, 2, 4, 6, or 8).
Why it works: Our number system is base-10, and only even numbers end in even digits.
Examples:
48 (ends with 8) Divisible
137 (ends with 7) Not Divisible
1,024 (ends with 4) Divisible
Divisibility by 3
Rule: A number is divisible by 3 if the sum of its digits is divisible by 3.
Why it works: In base-10, the remainder when dividing by 3 depends on the sum of digits.
Examples:
123: 1+2+3=6 (6÷3=2) Divisible
457: 4+5+7=16 (16÷3=5 R1) Not Divisible
891: 8+9+1=18 (18÷3=6) Divisible
Divisibility by 4
Rule: A number is divisible by 4 if its last two digits form a number divisible by 4.
Why it works: 100 is divisible by 4, so only the last two digits matter.
Examples:
316 (16÷4=4) Divisible
1,238 (38÷4=9.5) Not Divisible
2,024 (24÷4=6) Divisible
Divisibility by 5
Rule: A number is divisible by 5 if its last digit is 0 or 5.
Why it works: Multiples of 5 always end in 0 or 5 in our base-10 system.
Examples:
75 (ends with 5) Divisible
138 (ends with 8) Not Divisible
1,240 (ends with 0) Divisible
Basic Rules Practice
Explore practical applications and measure your knowledge using the divisibility-calculator.
Intermediate Divisibility Rules (6-10)
These rules build on the basic ones and are slightly more complex:
Divisibility by 6
Rule: A number is divisible by 6 if it is divisible by both 2 and 3.
Why it works: 6 = 2 × 3, and 2 and 3 are coprime (share no common factors).
Examples:
48: even (✓) and 4+8=12 (12÷3=4) Divisible
154: even (✓) but 1+5+4=10 (10÷3=3 R1) Not Divisible
237: 2+3+7=12 (✓) but odd (✗) Not Divisible
Divisibility by 7
Rule: Double the last digit and subtract it from the rest of the number. If the result is divisible by 7, then the original number is too.
Why it works: This rule comes from modular arithmetic properties of 7.
Examples:
161: 16 - (2×1) = 14 (14÷7=2) Divisible
245: 24 - (2×5) = 14 (14÷7=2) Divisible
328: 32 - (2×8) = 16 (16÷7=2 R2) Not Divisible
Divisibility by 8
Rule: A number is divisible by 8 if its last three digits form a number divisible by 8.
Why it works: 1,000 is divisible by 8, so only the last three digits matter.
Examples:
3,216 (216÷8=27) Divisible
7,430 (430÷8=53.75) Not Divisible
12,024 (24÷8=3) Divisible
Divisibility by 9
Rule: A number is divisible by 9 if the sum of its digits is divisible by 9.
Why it works: Similar to the rule for 3, but with a different modulus.
Examples:
378: 3+7+8=18 (18÷9=2) Divisible
457: 4+5+7=16 (16÷9=1 R7) Not Divisible
1,998: 1+9+9+8=27 (27÷9=3) Divisible
Divisibility by 10
Rule: A number is divisible by 10 if its last digit is 0.
Why it works: Multiples of 10 always end in 0 in our base-10 system.
Examples:
250 (ends with 0) Divisible
1,375 (ends with 5) Not Divisible
40,000 (ends with 0) Divisible
For larger numbers, you can use this alternative method:
- Group the digits in sets of three from right to left
- Add the groups in odd positions and subtract the groups in even positions
- If the result is divisible by 7, so is the original number
Example: 1,234,567
Groups: 1 | 234 | 567
Calculation: 567 - 234 + 1 = 334
334 ÷ 7 = 47.714... → Not divisible by 7
Advanced Divisibility Rules (11-13)
These rules are less commonly used but valuable for specific applications:
Divisibility by 11
Rule: Alternately add and subtract the digits from left to right. If the result is divisible by 11 (including 0), the number is divisible by 11.
Why it works: Based on the fact that 10 ≡ -1 (mod 11).
Examples:
121: 1-2+1=0 (0÷11=0) Divisible
2,915: 2-9+1-5=-11 (-11÷11=-1) Divisible
4,136: 4-1+3-6=0 (0÷11=0) Divisible
Divisibility by 12
Rule: A number is divisible by 12 if it is divisible by both 3 and 4.
Why it works: 12 = 3 × 4, and 3 and 4 are coprime.
Examples:
144: 1+4+4=9 (✓) and 44÷4=11 (✓) Divisible
156: 1+5+6=12 (✓) and 56÷4=14 (✓) Divisible
238: 2+3+8=13 (✗) though 38÷4=9.5 (✗) Not Divisible
Divisibility by 13
Rule: Multiply the last digit by 4 and add it to the rest of the number. If the result is divisible by 13, then the original number is too.
Why it works: Similar to the rule for 7, based on modular arithmetic.
Examples:
169: 16 + (4×9) = 52 (52÷13=4) Divisible
273: 27 + (4×3) = 39 (39÷13=3) Divisible
365: 36 + (4×5) = 56 (56÷13=4 R4) Not Divisible
Advanced Rules Practice
Put your learning to the test with real-case problems on the divisibility-calculator.
Divisibility Rules for Composite Numbers
For composite numbers (numbers with more than two factors), you can combine the rules for their prime factors:
Rule for Composite Numbers
A number is divisible by a composite number if it is divisible by all of its prime factors.
Example: Divisibility by 15
15 = 3 × 5
A number is divisible by 15 if it's divisible by both 3 and 5.
135: 1+3+5=9 (✓) and ends with 5 (✓) → Divisible by 15
Rule for Prime Powers
For numbers like 16, 25, 27, etc., you need to check divisibility by the prime raised to the appropriate power.
Example: Divisibility by 16
16 = 24
A number is divisible by 16 if its last 4 digits form a number divisible by 16.
1,232: 1232÷16=77 → Divisible by 16
Rule for Numbers with Repeated Factors
When a composite number has repeated prime factors, you need to ensure divisibility by the highest power of each prime.
Example: Divisibility by 18
18 = 2 × 32
A number is divisible by 18 if it's even and the sum of digits is divisible by 9.
234: even (✓) and 2+3+4=9 (✓) → Divisible by 18
| Number | Prime Factorization | Divisibility Rule |
|---|---|---|
| 6 | 2 × 3 | Divisible by 2 and 3 |
| 12 | 22 × 3 | Divisible by 3 and 4 |
| 14 | 2 × 7 | Divisible by 2 and 7 |
| 15 | 3 × 5 | Divisible by 3 and 5 |
| 18 | 2 × 32 | Divisible by 2 and 9 |
| 20 | 22 × 5 | Divisible by 4 and 5 |
| 21 | 3 × 7 | Divisible by 3 and 7 |
| 24 | 23 × 3 | Divisible by 3 and 8 |
Interactive Practice
Divisibility Checker
Test your knowledge by checking divisibility for any number and divisor combination.
Enter a number and divisor, then click "Check Divisibility"
Solution:
1. Take the last digit: 7
2. Double it: 7 × 2 = 14
3. Subtract from the rest of the number: 182 - 14 = 168
4. Check if 168 is divisible by 7: 16 - (2×8) = 16 - 16 = 0
5. Since 0 is divisible by 7, 168 is divisible by 7, so 1,827 is divisible by 7.
Answer: Yes, 1,827 is divisible by 7.
Solution:
1. Write the digits: 2, 4, 7, 5
2. Alternately add and subtract: 2 - 4 + 7 - 5 = 0
3. Since 0 is divisible by 11, 2,475 is divisible by 11.
Answer: Yes, 2,475 is divisible by 11.
Try out real-world exercises and test yourself with the divisibility-calculator.
Real-World Applications
Divisibility rules have practical applications in various fields:
Mathematics Education
Divisibility rules help students understand number properties, simplify fractions, and find factors quickly.
Example: Simplifying 24/36
Both divisible by 2, 3, 4, 6, 12
Greatest common factor is 12
24/36 = 2/3
Computer Science
Divisibility checks are used in algorithms, data structures, and optimization problems.
Example: Hash table implementation
Using modulo operations to distribute data
Quick checks for even/odd in bit manipulation
Finance and Accounting
Divisibility rules help with calculations involving percentages, ratios, and distributions.
Example: Splitting expenses
$120 split among 8 people: $15 each
Quick check: 120 divisible by 8? Last 3 digits: 120÷8=15 ✓
Everyday Life
From cooking measurements to event planning, divisibility rules simplify calculations.
Example: Party planning
36 guests, tables of 6: 36÷6=6 tables needed
Quick check: 36 divisible by 6? Even (✓) and 3+6=9 (✓)
Tips & Tricks
Master these strategies to become proficient with divisibility rules:
Memorize the Basic Rules First
Focus on rules for 2, 3, 5, and 10 initially
These are the most commonly used rules
Use Composite Rules
For numbers like 6, 12, 15, etc., check divisibility by their factors
This is often easier than memorizing separate rules
Practice Mental Math
Regular practice improves speed and accuracy
Try checking divisibility during daily activities
Understand Why Rules Work
Knowing the reasoning helps with recall
Makes it easier to apply rules correctly
- Confusing rules for 3 and 9: Remember that divisibility by 9 is stricter than divisibility by 3
- Misapplying the rule for 4: Always check the last two digits, not just the last digit
- Forgetting about 0: 0 is divisible by every number except itself
- Overcomplicating simple cases: Sometimes direct division is faster than applying complex rules
Quick Reference Guide
Use this table as a quick reference for all divisibility rules:
| Divisor | Rule | Example |
|---|---|---|
| 2 | Last digit is even (0, 2, 4, 6, 8) | 48 (ends with 8) ✓ |
| 3 | Sum of digits divisible by 3 | 123 (1+2+3=6) ✓ |
| 4 | Last two digits divisible by 4 | 316 (16÷4=4) ✓ |
| 5 | Last digit is 0 or 5 | 75 (ends with 5) ✓ |
| 6 | Divisible by both 2 and 3 | 48 (even and 4+8=12) ✓ |
| 7 | Double last digit, subtract from rest | 161 (16-2=14) ✓ |
| 8 | Last three digits divisible by 8 | 3,216 (216÷8=27) ✓ |
| 9 | Sum of digits divisible by 9 | 378 (3+7+8=18) ✓ |
| 10 | Last digit is 0 | 250 (ends with 0) ✓ |
| 11 | Alternating sum divisible by 11 | 121 (1-2+1=0) ✓ |
| 12 | Divisible by both 3 and 4 | 144 (1+4+4=9 and 44÷4=11) ✓ |
| 13 | Multiply last digit by 4, add to rest | 169 (16+36=52) ✓ |