Divisibility Test Calculator
Test if numbers are divisible by 2-13 using mathematical rules and step-by-step explanations
Test Divisibility
Enter any positive integer to test its divisibility
Quick Reference
Example Tests
Example 1: Testing 144
Example 2: Testing 1001
Example 3: Testing 123
Divisibility Rules
Quick Tips
Start with easy tests like 2, 5, and 10
For composite numbers, test their prime factors
Sum-based rules (3, 9) can be applied repeatedly
Practice alternating sums for 11 and 13
Remember: 0 is divisible by all numbers
Understanding Divisibility Tests
What are Divisibility Tests?
Divisibility tests are mathematical shortcuts that help determine whether one number divides another without performing the actual division. These rules save time and make mental math easier, especially with large numbers.
Why are They Important?
- •Speed up mental arithmetic calculations
- •Help in factorization and simplifying fractions
- •Useful in number theory and cryptography
- •Build understanding of number patterns
Types of Rules
Last Digit Rules
For 2, 5, 10: Check only the last digit(s)
Sum Rules
For 3, 9: Add all digits and test the sum
Alternating Rules
For 11, 13: Alternating addition and subtraction
Composite Rules
For 6, 12: Test divisibility by prime factors
Mathematical Foundation
Most divisibility rules work because of modular arithmetic properties. For example, since 10 ≡ 1 (mod 3), each digit position contributes equally to the remainder when dividing by 3, making the sum-of-digits rule valid.
Powers of 10
10¹ ≡ 1 (mod 3, 9)
10² ≡ 1 (mod 4)
10³ ≡ 1 (mod 8)
Alternating Patterns
10 ≡ -1 (mod 11)
10³ ≡ -1 (mod 13)
Creating alternating sums