Divisor Calculator
Find all divisors (factors) of any integer or common divisors of two numbers
Find Divisors
Divisors of 24
All Divisors
Prime Factorization
Divisibility Rules Applied
Example Problems
Example 1: Divisors of 24
Problem: Find all divisors of 24.
Solution:
• Check: 1 ✓ (24 ÷ 1 = 24)
• Check: 2 ✓ (24 ÷ 2 = 12, last digit 4 is even)
• Check: 3 ✓ (24 ÷ 3 = 8, sum of digits: 2+4=6, divisible by 3)
• Check: 4 ✓ (24 ÷ 4 = 6)
• Check: 6 ✓ (24 ÷ 6 = 4, divisible by both 2 and 3)
• Check: 8 ✓ (24 ÷ 8 = 3)
• Check: 12 ✓ (24 ÷ 12 = 2)
• Check: 24 ✓ (24 ÷ 24 = 1)
Answer: 1, 2, 3, 4, 6, 8, 12, 24 (8 divisors)
Example 2: Divisors of 36
Problem: Find all divisors of 36.
Solution:
• Prime factorization: 36 = 2² × 3²
• Using systematic checking up to √36 = 6:
• Divisors: 1, 2, 3, 4, 6, 9, 12, 18, 36
• Proper divisors: 1, 2, 3, 4, 6, 9, 12, 18
Answer: 9 total divisors
Example 3: Common Divisors of 24 and 36
Problem: Find common divisors of 24 and 36.
Solution:
• Divisors of 24: 1, 2, 3, 4, 6, 8, 12, 24
• Divisors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
• Common divisors: 1, 2, 3, 4, 6, 12
• Greatest Common Divisor (GCD): 12
Answer: 6 common divisors
Divisibility Rules
Quick Rules
• 2: Last digit is even
• 3: Sum of digits ÷ 3
• 4: Last two digits ÷ 4
• 5: Ends in 0 or 5
• 6: Divisible by 2 and 3
• 8: Last three digits ÷ 8
• 9: Sum of digits ÷ 9
• 10: Ends in 0
Key Concepts
Divisor
A number that divides evenly into another
Factor
Same as divisor, used interchangeably
Proper Divisor
All divisors except the number itself
GCD
Greatest Common Divisor of two numbers
Understanding Divisors and Factors
What are Divisors?
A divisor (or factor) of an integer n is a number that divides n evenly, meaning the division results in an integer with no remainder. For example, the divisors of 12 are 1, 2, 3, 4, 6, and 12 because each of these numbers divides 12 evenly.
Finding Divisors
To find all divisors of a number n, we only need to check numbers up to √n. For each divisor d we find, n/d is also a divisor. This method is much more efficient than checking every number from 1 to n.
Applications
- •Number theory and prime factorization
- •Greatest Common Divisor (GCD) calculations
- •Simplifying fractions and ratios
- •Cryptography and computer science
Mathematical Properties
Divisibility Rules
Quick methods to check if a number is divisible by small integers:
- • Even numbers are divisible by 2
- • If digit sum is divisible by 3, so is the number
- • Numbers ending in 0 or 5 are divisible by 5
Prime Factorization
Every positive integer can be expressed as a product of prime numbers. This factorization helps us find all divisors systematically.
Number of Divisors
If n = p₁^a₁ × p₂^a₂ × ... × pₖ^aₖ, then the number of divisors is (a₁ + 1) × (a₂ + 1) × ... × (aₖ + 1).