Divisor - Definition, Examples, Quiz, FAQ, Trivia
Learn about divisors with simple explanations, visual examples, and practice activities
What is a Divisor?
A divisor is a number that divides another number exactly without leaving a remainder.
For example, the divisors of 12 are 1, 2, 3, 4, 6, and 12 because:
12 ÷ 1 = 12, 12 ÷ 2 = 6, 12 ÷ 3 = 4, 12 ÷ 4 = 3, 12 ÷ 6 = 2, and 12 ÷ 12 = 1.
Divisors are the building blocks of numbers and help us understand how numbers are made.
Every number has at least two divisors: 1 and itself.
Key Concept
Divisors are like friends that share a number equally without leftovers!
Finding Divisors
To find all divisors of a number, we look for all numbers that divide it evenly. Here are important concepts:
Divisor Pairs
Divisors come in pairs that multiply to the original number. For 18:
1 × 18 = 18, 2 × 9 = 18, 3 × 6 = 18
So the divisor pairs are (1,18), (2,9), and (3,6).
Prime Divisors
Prime divisors are divisors that are prime numbers (only divisible by 1 and themselves). For 30, the divisors are 1, 2, 3, 5, 6, 10, 15, 30. The prime divisors are 2, 3, and 5.
Prime Factorization
This is breaking a number down into its prime building blocks. For 36:
36 = 2 × 2 × 3 × 3 = 2² × 3²
Divisor Count Formula
To find how many divisors a number has:
- Find the prime factorization (e.g., 36 = 2² × 3²)
- Add 1 to each exponent (2+1=3, 2+1=3)
- Multiply these numbers: 3 × 3 = 9 divisors
Remember
You only need to check numbers up to the square root when finding divisors!
Methods to Find Divisors
There are several ways to find divisors:
Trial Division
This is the simplest method where we test all numbers from 1 up to the number itself (or the square root) to see which ones divide evenly.
Example for 20:
20 ÷ 1 = 20 ✔, 20 ÷ 2 = 10 ✔, 20 ÷ 3 = 6.66 ✘, 20 ÷ 4 = 5 ✔, ... up to 20 ÷ 20 = 1 ✔
Euclidean Algorithm
This method helps find the Greatest Common Divisor (GCD) of two numbers, which is the largest number that divides both.
Steps to find GCD of 48 and 18:
48 ÷ 18 = 2 remainder 12
18 ÷ 12 = 1 remainder 6
12 ÷ 6 = 2 remainder 0 → GCD is 6
Pro Tip
For large numbers, prime factorization is often the fastest way to find all divisors!
The Divisor Game
The divisor game is a math activity where two players take turns:
- Start with a number (e.g., 30)
- Player 1 chooses a proper divisor (a divisor other than the number itself)
- Divide the number by that divisor to get a new number
- Player 2 then chooses a proper divisor of the new number
- The player who reaches 1 wins!
Example Game
Starting with 30:
- Player 1 chooses 5 → 30 ÷ 5 = 6
- Player 2 chooses 3 → 6 ÷ 3 = 2
- Player 1 chooses 2 → 2 ÷ 2 = 1 (Player 1 wins!)
Strategy Tip
Choosing prime divisors often leads to quicker wins!
Practice Quiz
Test your understanding of divisors with these questions:
Frequently Asked Questions
Here are answers to common questions about divisors:
Math Trivia
Discover interesting facts about divisors and numbers:
Ancient Number Theory
Ancient Greek mathematicians studied divisors over 2,000 years ago. Euclid's "Elements" written around 300 BC contains important theorems about divisors.
Prime Number Secrets
Prime numbers have exactly two divisors: 1 and themselves. The largest known prime number has over 24 million digits!
Divisors in Security
Finding divisors of very large numbers is crucial for modern internet security. This difficulty protects your online information!
Highly Composite Numbers
1 is the only number with one divisor. 60 has 12 divisors - more than any smaller number, making it "highly composite".
