Factors and Multiples
Learn what factors and multiples are, how to find them, and why they're important in math.
Definition
- (no remainder)
- (no remainder)
- (no remainder)
- (no remainder)
- (no remainder)
- (no remainder)
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Worked Examples
Find all the factors of 18.
Start with 1
, so 1 and 18 are factors → Factors:
Try 2
, so 2 and 9 are factors → Factors:
Try 3
, so 3 and 6 are factors → Factors:
Try 4 and 5
(not whole), (not whole) → 4 and 5 are NOT factors
Stop at the square root
, so we've found all factors → All factors:
Answer: The factors of 18 are:
Common Mistakes
Confusing factors and multiples
Why it's wrong: The words sound similar but mean opposite things. Factors divide INTO a number; multiples are made BY multiplying.
Correct: Remember: Factors are Few (limited), Multiples are Many (infinite). 12 has 6 factors, but infinite multiples.
Forgetting that 1 is always a factor
Why it's wrong: Every number divided by 1 gives itself, so 1 is always a factor.
Correct: Always start your factor list with 1, and don't forget the number itself is also a factor.
Thinking 0 is a multiple of every number
Why it's wrong: While is true, we usually start listing multiples from the number itself.
Correct: The first multiple of any number is itself (not 0). So multiples of 5 start:
Missing factor pairs
Why it's wrong: When finding factors, students sometimes forget that factors come in pairs.
Correct: When you find one factor, find its pair: if , then both 2 AND 9 are factors.
Interactive Visual
Factor Tree
Enter a number to see its factor tree and prime factorization.
Venn Diagram - GCF & LCM
Prime factors of 12:
Prime factors of 18:
GCF
6
2 × 3 = 6
LCM
36
2 × 2 × 3 × 3 = 36
How it works:
- • GCF = product of common prime factors (center of diagram)
- • LCM = product of all prime factors (entire diagram)
Explore the factors of two numbers.
Interactive Sandbox
Expression Calculator
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Practice Problems
16 problemsWhich number is a factor of 20?
Why It Matters
- Simplifying fractions: Find common factors to reduce to
- Adding fractions: Find common multiples to get the same denominator
- Sharing equally: Divide 24 cookies among 6 friends (24 is divisible by 6)
- Patterns: Understand why every even number has 2 as a factor
- Problem solving: Figure out schedules, arrangements, and groupings
Real World Applications
Party Planning
When organizing events, you need to divide things equally among guests.
Example:
You have 30 cupcakes. Which group sizes can share them equally? The factors of 30: . So you can have groups of 2, 3, 5, 6, 10, 15, or 30 people.
You're planning a party with 24 party favors to distribute equally.
How many guests can you invite so everyone gets the same number of favors?
Step 1: Write the mathematical expression
Find the factors of 24:
Scheduling and Patterns
Multiples help us find when events occur together.
Example:
Bus A comes every 15 minutes. Bus B comes every 20 minutes. When do they arrive together? Find common multiples of 15 and 20: They arrive together every 60 minutes.
You practice piano every 3 days and soccer every 4 days.
If you have both today, in how many days will you have both again?
Step 1: Write the mathematical expression
Find the first common multiple of 3 and 4:
Arranging Objects
Factors help determine how to arrange items in rows and columns.
Example:
You have 48 chairs to arrange. Possible arrangements: 1 row of 48, 2 rows of 24, 3 rows of 16, 4 rows of 12, 6 rows of 8, 8 rows of 6, etc.
A classroom has 28 desks to arrange in equal rows.
What are all the possible arrangements?
Step 1: Write the mathematical expression
Find factor pairs of 28:
Key Takeaways
- 1Factors divide evenly into a number with no remainder
- 2Multiples are made by multiplying a number by 1, 2, 3, ...
- 3Every number has 1 and itself as factors
- 4Factors come in pairs that multiply to give the original number
- 5Multiples are infinite, but factors are finite
- 6Factors help with division and simplifying; multiples help with patterns and scheduling
Frequently Asked Questions
Glossary
- Factor
- A number that divides evenly into another number with no remainder
- Multiple
- The product of a number and any whole number
- Factor pair
- Two numbers that multiply together to give a specific product
- Divisible
- A number is divisible by another if the division has no remainder
- Prime number
- A number with exactly two factors: 1 and itself