H.2 Identify factors of a given number
What are factors?
Factors are numbers we can multiply together to get another number. A factor divides evenly into a whole number, leaving no remainder. Think of factors as the “building blocks” that multiply to make a product.
- 3 × 4 = 12, so 3 and 4 are factors of 12.
- 2 × 6 = 12, so 2 and 6 are also factors of 12.
- 1 × 12 = 12, so 1 and 12 are also factors of 12.
A factor always pairs with another factor. When you divide a number by one of its factors, the quotient is the other factor. For example, 12 ÷ 3 = 4, so 3 and 4 are a factor pair.
Factor pairs
A factor pair is two whole numbers that multiply together to give a specific product. Finding factor pairs helps us list all the factors of a number in an organized way.
- For 18: (1, 18), (2, 9), (3, 6) — these are all the factor pairs.
- For 24: (1, 24), (2, 12), (3, 8), (4, 6).
- For 7: (1, 7) — the only factor pair, because 7 is a prime number.
When listing factor pairs, we usually start with 1 and the number itself, then check 2, 3, 4, and so on until the numbers in the pair get closer together or meet.
How to find all factors of a number
To find every factor of a whole number, test each smaller number (starting from 1) to see if it divides the target number evenly. If it does, both the divisor and the quotient are factors. Stop when you reach a number that has already appeared as a factor.
- 1 × 36 → factors 1 and 36
- 2 × 18 → factors 2 and 18
- 3 × 12 → factors 3 and 12
- 4 × 9 → factors 4 and 9
- 5 does not divide 36 evenly → skip
- 6 × 6 → factors 6 and 6 (only list 6 once)
- Next number would be 7, but 7 × 7 is greater than 36, and we already have all pairs.
All factors of 36 are: 1, 2, 3, 4, 6, 9, 12, 18, 36.
If a number does not divide evenly (there is a remainder), it is not a factor. Use multiplication facts you know to speed up the process.
Factors vs. multiples
Sometimes students confuse factors and multiples. Remember: factors are what you multiply to get a number; multiples are what you get after multiplying a number by 1, 2, 3, and so on.
- Factors of 12: 1, 2, 3, 4, 6, 12 (all numbers that divide 12 evenly).
- Multiples of 12: 12, 24, 36, 48, … (12 × 1, 12 × 2, 12 × 3 …).
- Think: Factors are smaller than or equal to the number; multiples are larger than or equal to the number.
A helpful trick: “Factors are finite; multiples are infinite.” A number has a limited set of factors, but it has an endless list of multiples.
Common factors
A common factor is a number that divides two or more whole numbers exactly. It is a factor that appears in the factor lists of both numbers.
- Factors of 15: 1, 3, 5, 15.
- Factors of 20: 1, 2, 4, 5, 10, 20.
- Common factors of 15 and 20: 1 and 5. (Both 1 and 5 appear in both lists.)
The greatest common factor (GCF) is the largest number in the list of common factors. For 15 and 20, the GCF is 5.
Prime and composite numbers
A prime number has exactly two distinct factors: 1 and itself. A composite number has more than two factors. The number 1 is neither prime nor composite; it has only one factor (1).
- Prime numbers: 2 (factors 1, 2); 7 (factors 1, 7); 13 (factors 1, 13).
- Composite numbers: 4 (factors 1, 2, 4); 10 (factors 1, 2, 5, 10); 25 (factors 1, 5, 25).
- 1 is special — it has only one factor, so it is neither prime nor composite.
Every whole number greater than 1 is either prime or composite. Knowing factors helps you decide which is which.
Why do we learn factors?
Understanding factors helps with simplifying fractions, solving division problems, finding patterns in numbers, and preparing for multiplication and division of larger numbers. Factors are everywhere in math!
- If you have 24 cookies and want to share them equally among friends, the number of friends must be a factor of 24 (1, 2, 3, 4, 6, 8, 12, or 24).
- Arranging chairs in rows: If you have 30 chairs, possible row sizes are factors of 30.
Factors help us break numbers into smaller parts. This skill is the foundation for many math topics, including fractions and division.
Tips for finding factors quickly
Use divisibility rules to check possible factors without doing long division. This makes finding factors faster and more accurate.
- A number is divisible by 2 if it ends in 0, 2, 4, 6, or 8.
- Divisible by 3 if the sum of its digits is divisible by 3.
- Divisible by 4 if the last two digits form a number divisible by 4.
- Divisible by 5 if it ends in 0 or 5.
- Divisible by 6 if it is divisible by both 2 and 3.
- Divisible by 9 if the sum of its digits is divisible by 9.
- Divisible by 10 if it ends in 0.
Practice these rules. They are like shortcuts that save time and help you avoid missing factors.
Examples of factor lists for numbers up to 100
Here are some more factor lists to study. Notice how prime numbers have only two factors, while composite numbers have more.
- 16: 1, 2, 4, 8, 16
- 21: 1, 3, 7, 21
- 28: 1, 2, 4, 7, 14, 28
- 32: 1, 2, 4, 8, 16, 32
- 45: 1, 3, 5, 9, 15, 45
- 49: 1, 7, 49 (49 is 7×7, a square number)
- 56: 1, 2, 4, 7, 8, 14, 28, 56
- 64: 1, 2, 4, 8, 16, 32, 64
- 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72
When a number is a perfect square (like 36 or 49), one factor pair will have two identical numbers (6 × 6 or 7 × 7). List that factor only once.
Special cases: zero and one
Zero and one have unique factor properties. Zero is divisible by any non-zero number, but we usually do not list factors of zero. One has exactly one factor: itself.
- 1 × 1 = 1, so the only factor of 1 is 1.
- 0 ÷ 5 = 0 with no remainder, so 5 is a factor of 0. In fact, every whole number except 0 is a factor of 0. Because of this, mathematicians usually do not talk about "factors of zero" in elementary school.
For fourth grade, focus on finding factors of whole numbers greater than 1. This is where factor pairs are most useful.
Using factors to solve problems
Factors are not just a list—they help us solve real math problems. For example, if you need to arrange 48 desks into equal rows, the number of rows must be a factor of 48.
- Maria has 30 stickers. She wants to put them into albums so that each page has the same number of stickers and no stickers are left over. What are the possible numbers of stickers per page?
- Solution: Find all factors of 30: 1, 2, 3, 5, 6, 10, 15, 30. Maria can put 1, 2, 3, 5, 6, 10, 15, or 30 stickers on each page.
When solving problems, ask yourself: "Does this number divide evenly?" If yes, it's a factor and can be used to make equal groups.
Common Core alignment: CCSS.MATH.CONTENT.4.OA.B.4 – Find all factor pairs for a whole number in the range 1-100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1-100 is prime or composite.
Notes for teachers
This lesson is aligned with CCSS.MATH.CONTENT.4.OA.B.4. All content is 100% free; use it for whole-class instruction, independent study and practice, or homework.