What is the rule for divisibility by 7?

Double the last digit and subtract it from the rest of the number. Repeat if needed.

Divisibility Calculator with Step-by-Step Solutions

What is Divisibility?

Divisibility is a mathematical concept that determines whether one number can be divided by another without leaving a remainder. Divisibility rules are shortcuts that help quickly determine this without performing complete division.

Key Concepts:

Divisor: The number by which we divide
Dividend: The number being divided
Quotient: The result of division
Remainder: The amount left over after division
Divisibility Rule: A shortcut to determine divisibility

Basic Divisibility

A number is divisible by another if the division results in a whole number with no remainder.

15 ÷ 3 = 5 ✓
15 ÷ 4 = 3.75 ✗
15 is divisible by 3 but not by 4

Importance of Rules

Divisibility rules save time and help in mental math, factorization, and problem-solving.

Instead of dividing 123 by 3:
Sum digits: 1+2+3=6
6 ÷ 3 = 2 ✓
∴ 123 is divisible by 3

Common Uses

Divisibility rules are used in simplifying fractions, finding factors, and checking mathematical properties.

Simplify 24/36:
Both divisible by 12
24÷12=2, 36÷12=3
Simplified: 2/3

Common Divisibility Rules

These are the most frequently used divisibility rules for numbers 2-20:

Divisibility by 2

A number is divisible by 2 if its last digit is even (0, 2, 4, 6, or 8).

Examples:
24 → last digit 4 (even) ✓
37 → last digit 7 (odd) ✗

Divisibility by 3

A number is divisible by 3 if the sum of its digits is divisible by 3.

123 → 1+2+3=6 ✓
6 ÷ 3 = 2 ✓
∴ 123 is divisible by 3

Divisibility by 4

A number is divisible by 4 if its last two digits form a number divisible by 4.

1324 → last two digits 24 ✓
24 ÷ 4 = 6 ✓
∴ 1324 is divisible by 4

Divisibility by 5

A number is divisible by 5 if its last digit is 0 or 5.

125 → last digit 5 ✓
238 → last digit 8 ✗
∴ 125 divisible by 5

Divisibility by 6

A number is divisible by 6 if it is divisible by both 2 and 3.

24 → even ✓
2+4=6 ÷ 3=2 ✓
∴ 24 divisible by 6

Divisibility by 9

A number is divisible by 9 if the sum of its digits is divisible by 9.

729 → 7+2+9=18 ✓
18 ÷ 9 = 2 ✓
∴ 729 divisible by 9

Advanced Divisibility Rules

These rules are less common but useful for specific divisors:

Divisibility by 7

Double the last digit and subtract from the rest. Repeat if necessary.

182 → 18 - (2×2) = 14 ✓
14 ÷ 7 = 2 ✓
∴ 182 divisible by 7

Divisibility by 8

A number is divisible by 8 if its last three digits form a number divisible by 8.

3128 → last three digits 128 ✓
128 ÷ 8 = 16 ✓
∴ 3128 divisible by 8

Divisibility by 11

Alternate sum of digits (sum of odd positions - sum of even positions) must be divisible by 11.

121 → (1+1) - 2 = 0 ✓
0 ÷ 11 = 0 ✓
∴ 121 divisible by 11

Divisibility by 13

Multiply last digit by 4 and add to remaining number. Repeat if necessary.

169 → 16 + (9×4) = 52 ✓
52 ÷ 13 = 4 ✓
∴ 169 divisible by 13

Divisibility by 17

Multiply last digit by 5 and subtract from remaining number. Repeat if necessary.

153 → 15 - (3×5) = 0 ✓
0 ÷ 17 = 0 ✓
∴ 153 divisible by 17

Divisibility by 19

Multiply last digit by 2 and add to remaining number. Repeat if necessary.

133 → 13 + (3×2) = 19 ✓
19 ÷ 19 = 1 ✓
∴ 133 divisible by 19

Real-World Applications of Divisibility Rules

Divisibility rules have numerous practical applications in everyday life and various fields:

Mathematics Education

Simplifying fractions and ratios
Finding factors and multiples
Mental math calculations
Number theory problems

Computer Science

Algorithm optimization
Data validation
Hash function design
Error detection codes

Finance & Accounting

Check digit verification
Account number validation
Tax calculation shortcuts
Financial ratio analysis

Engineering

Measurement conversions
Component sizing
Grid system design
Optimization problems

Daily Life

Quick mental calculations
Budgeting and shopping
Recipe scaling
Time management

Cryptography

Prime number testing
Key generation
Encryption algorithms
Security protocols

Solved Divisibility Examples

Step-by-step solutions to common divisibility problems:

Example 1: Check divisibility of 126 by 2, 3, 6, 9

Determine which numbers divide 126 evenly.

1. By 2: Last digit 6 (even) ✓

2. By 3: 1+2+6=9 ÷ 3=3 ✓

3. By 6: Divisible by 2 and 3 ✓

4. By 9: 1+2+6=9 ÷ 9=1 ✓

Result: Divisible by 2, 3, 6, and 9

Example 2: Check divisibility of 245 by 5 and 7

Test if 245 is divisible by 5 and 7.

1. By 5: Last digit 5 ✓

2. By 7: 24 - (5×2) = 14 ✓

3. 14 ÷ 7 = 2 ✓

Result: Divisible by both 5 and 7

Example 3: Check divisibility of 1320 by 4, 8, 11

Determine divisibility of 1320 by 4, 8, and 11.

1. By 4: Last two digits 20 ÷ 4=5 ✓

2. By 8: Last three digits 320 ÷ 8=40 ✓

3. By 11: (1+2) - (3+0) = 0 ✓

Result: Divisible by 4, 8, and 11

Example 4: Find all divisors of 72

Use divisibility rules to find all factors of 72.

1. Test divisors 1-12

2. Apply divisibility rules

3. Verify with division

4. Factors: 1,2,3,4,6,8,9,12,18,24,36,72

Result: 12 divisors total

Example 5: Check divisibility of 1001 by 7, 11, 13

Test the famous number 1001 for divisibility by 7, 11, and 13.

1. By 7: 100 - (1×2) = 98 ✓

2. 98 ÷ 7 = 14 ✓

3. By 11: (1+0) - (0+1) = 0 ✓

4. By 13: 100 + (1×4) = 104 ✓

Result: Divisible by 7, 11, and 13

Example 6: Simplify fraction using divisibility

Simplify 84/126 using divisibility rules.

1. Find GCD using divisibility

2. Both divisible by 2, 3, 6, 7, 14, 21, 42

3. Greatest common divisor: 42

4. 84÷42=2, 126÷42=3

Result: Simplified to 2/3

Practice Problems

Test your understanding with these divisibility problems:

Problem 1: Is 357 divisible by 7? Show your work.

Solution:

Using divisibility rule for 7:

35 - (7×2) = 35 - 14 = 21

21 ÷ 7 = 3 ✓

Therefore, 357 is divisible by 7.

Problem 2: Find all numbers between 1-50 that are divisible by both 3 and 4.

Solution:

Numbers divisible by both 3 and 4 are divisible by 12 (LCM of 3 and 4).

12, 24, 36, 48

Total: 4 numbers

Problem 3: Check if 123456 is divisible by 8.

Solution:

Divisibility rule for 8: Check last three digits.

Last three digits: 456

456 ÷ 8 = 57 ✓

Therefore, 123456 is divisible by 8.

Problem 4: Is 10101 divisible by 37? Use divisibility rule for 37.

Solution:

Rule for 37: Group digits in sets of three from right.

10 | 101

10 + 101 = 111

111 ÷ 37 = 3 ✓

Therefore, 10101 is divisible by 37.

Problem 5: Simplify 294/378 using divisibility rules.

Solution:

Find GCD using divisibility:

Both divisible by 2, 3, 6, 7, 14, 21, 42

Greatest common divisor: 42

294 ÷ 42 = 7, 378 ÷ 42 = 9

Simplified: 7/9

How to Use Divisibility Rules Step-by-Step

Follow this systematic approach to check divisibility efficiently:

1

Identify the Divisor

Determine which divisibility rule to apply based on the divisor.

Divisor: 6
Rules needed: 2 and 3

2

Apply the Rule

Use the appropriate divisibility rule for the specific divisor.

For divisor 3:
Sum digits: 1+2+3=6
6 ÷ 3 = 2 ✓

3

Check Multiple Rules

For composite divisors, check divisibility by all prime factors.

Divisor 12:
Check 3 and 4
Both must be satisfied

4

Verify with Division

When in doubt, perform actual division to confirm.

126 ÷ 6 = 21 ✓
Confirms divisibility

5

Record the Result

Note whether the number is divisible or not.

126: Divisible by 2,3,6,7,9,14,18,21,42,63,126

6

Apply to Problems

Use results for simplification, factorization, or problem-solving.

Simplify fractions
Find factors
Solve word problems

Pro Tips for Divisibility Tests

Memorize common rules: 2,3,5,9,10 are most frequently used
Combine rules: For composite numbers, check all prime factors
Practice mental math: Improves speed and accuracy
Use shortcuts: Last digit patterns help with many divisors
Verify with calculator: When unsure, double-check with division

Divisibility Rules – Frequently Asked Questions

Learn how divisibility rules work, how to check numbers quickly, and how to use them in real math problems.

Why do divisibility rules work?

Divisibility rules work because of number properties in the base-10 system. For example, divisibility by 3 depends on the sum of digits because 10 leaves a remainder of 1 when divided by 3.

Are there divisibility rules for all numbers?

Yes, but some are complex. Simple rules exist for numbers like 2, 3, 5, and 10, while larger or prime numbers often require advanced methods.

What is the easiest divisibility rule?

The easiest rule is for 2: a number is divisible by 2 if its last digit is even (0, 2, 4, 6, 8).

How do you check divisibility by 3?

Add all digits of the number. If the sum is divisible by 3, then the number is also divisible by 3.

What is the divisibility rule for 7?

Double the last digit and subtract it from the remaining number. Repeat the process until you get a smaller number to test.

Can I create my own divisibility rules?

Yes, divisibility rules can be derived mathematically using modular arithmetic, though some may become complex.

Do divisibility rules work for large numbers?

Yes, they work for any size number and are especially useful for simplifying large calculations quickly.

What is a divisible number?

A number is divisible by another if it can be divided without leaving a remainder.

What is the rule for divisibility by 9?

A number is divisible by 9 if the sum of its digits is divisible by 9.

Why are divisibility rules important?

They help simplify calculations, check answers quickly, and are widely used in algebra and number theory.

Are divisibility rules used in real life?

Yes, they are used in computing, cryptography, financial calculations, and quick mental math.

Do divisibility rules change in different number systems?

Yes, rules depend on the base of the number system, so they differ in binary, hexadecimal, and others.

Is this divisibility calculator accurate?

Yes, it uses precise mathematical algorithms to provide accurate and instant results for all numbers.

Related Mathematical Calculators

Explore our collection of number theory and mathematics tools:

Related Number Theory Learning Guides

Strengthen your understanding with these in-depth number theory tutorials and essential concepts:

Related Number Theory Learning Guides

Explore essential number theory concepts with clear explanations, practical examples, and step-by-step problem-solving techniques.

Divisibility Rules Calculator (2–100+) with Step-by-Step Solutions