Common Factors - Definition, Examples, Quiz, FAQ, Trivia
Learn how to find common factors with easy explanations, examples, and practice activities
What Are Common Factors?
Common factors are numbers that divide exactly into two or more other numbers. They're like shared building blocks that different numbers have in common.
For example, the numbers 12 and 18 share some factors:
• Factors of 12: 1, 2, 3, 4, 6, 12
• Factors of 18: 1, 2, 3, 6, 9, 18
The common factors of 12 and 18 are the numbers that appear in both lists: 1, 2, 3, and 6.
Understanding common factors helps us simplify fractions, solve problems about sharing equally, and find patterns in numbers.
Key Concept
Every pair of numbers has at least one common factor: the number 1. This is because 1 divides evenly into every whole number.
Finding Common Factors
Here's how to find common factors of two numbers:
Step 1: Find all factors of the first number
Step 2: Find all factors of the second number
Step 3: Identify which numbers appear in both lists
Let's practice with 24 and 36:
Finding Factors
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
Common factors: 1, 2, 3, 4, 6, 12
Remember: Always start with 1 and the number itself when listing factors!
Remember
Factors are always whole numbers that divide evenly (no remainders). The number of factors a number has can vary - prime numbers only have two!
Greatest Common Factor (GCF)
The Greatest Common Factor (GCF), also called the Highest Common Factor (HCF), is the largest number that divides exactly into two or more numbers.
Why is GCF important? It helps us:
• Simplify fractions to their lowest terms
• Solve problems about equal distribution
• Understand number relationships
Finding GCF
Method 1: List all factors and choose the largest common one
Method 2: Use prime factorization (breaking numbers into prime factors)
• Factors of 28: 1, 2, 4, 7, 14, 28
• Factors of 42: 1, 2, 3, 6, 7, 14, 21, 42
• Common factors: 1, 2, 7, 14
• GCF is 14
Pro Tip
When numbers have a GCF of 1, we say they are "coprime" or "relatively prime." This means they share no common factors other than 1.
Real-World Examples
Common factors help solve many real problems. Here are some examples:
Example 1: Sharing cookies
You have 12 chocolate chip cookies and 18 oatmeal cookies. What's the largest number of friends you can share them with equally?
Solution: Find GCF of 12 and 18 → 6 friends
Example 2: Arranging plants
A gardener has 24 roses and 36 tulips. She wants to arrange them in equal rows with only one type in each row. What's the greatest number of plants per row?
Solution: GCF of 24 and 36 → 12 plants per row
Example 3: Simplifying fractions
Simplify 16/24 to lowest terms:
GCF of 16 and 24 is 8, so divide both by 8 → 2/3
Example 4: Party planning
You have 20 balloons and 30 streamers. How many identical party packs can you make using all supplies?
Solution: GCF of 20 and 30 → 10 party packs
Practice Tip
Look for common factors in your daily life - when dividing snacks, organizing items, or sharing toys with friends!
Common Factors Quiz
Test your understanding with this 5-question quiz. Choose the correct answer for each question.
Frequently Asked Questions
Here are answers to common questions about common factors:
Math Trivia
Discover interesting facts about numbers and factors:
Ancient Number Theory
The concept of common factors dates back to ancient Greek mathematicians like Euclid, who described an algorithm for finding GCF around 300 BCE. This method, called the Euclidean algorithm, is still used today!
Perfect Numbers
A perfect number equals the sum of its proper factors (factors excluding itself). 6 is perfect (1+2+3=6), and so is 28 (1+2+4+7+14=28). All known perfect numbers are even - mathematicians still don't know if odd perfect numbers exist!
Prime Numbers
Prime numbers have exactly two factors: 1 and themselves. The largest known prime number (as of 2023) has over 24 million digits! Primes are crucial for internet security through encryption.
Factor Records
The number with the most factors under 100 is 60 (12 factors: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60). Under 1000, it's 840 with 32 factors!
