What Are the Factors of 36? Math Guide with 4 Method

Factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36 while pair factors of 36 are (1, 36), (2, 18), (3, 12), (4, 9) and (6, 6). A factor of a number is an integer that divides it without a remainder. Factors of 36 are numbers that can divide the number 36 completely.

We can find factors of 36 by performing prime factorization or integer factorization of a number is breaking a number down into a set of prime numbers whose product results in the original number.

• Positive pair factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36
• Negative pair factors of 36 are -1, -2, -3, -4, -6, -9, -12, -18 and -36
• Prime Factors of 36 are 2 and 3
• Sum of Factors of 36 is 91
• Prime Factorization of 36 is 2 × 2 × 3 × 3

What Are the Factors of 36?

To find all factors of 36:

1. List pairs of numbers that multiply to 36.
2. Check divisibility starting from 1 to 36.

Result:
The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, 36

All the Factors of 36

Positive Factors

The positive factors of 36 are found by testing divisibility or using prime factorization:
1, 2, 3, 4, 6, 9, 12, 18, 36.

Negative Factors

Negative factors of 36 are simply the negatives of its positive factors:
-1, -2, -3, -4, -6, -9, -12, -18, -36.

Finding Factors via Divisibility Rules

Use these shortcuts to identify factors:

• Divisible by 2: Even numbers (e.g., 2, 4, 6).
• Divisible by 3: Sum of digits is divisible by 3 (3 + 6 = 9 → divisible by 3).
• Divisible by 4: Last two digits form a number divisible by 4 (36 → 36 ÷ 4 = 9).
• Divisible by 6: Divisible by both 2 and 3.
• Divisible by 9: Sum of digits is divisible by 9.

Example:
Testing 12 as a factor:
36 ÷ 12 = 3 → No remainder.
Thus, 12 is a factor.

3 Easy Ways to Find the Factors of 36

Method 1: The Factor Pair Method (Rainbow Method)

This is the most visual and beginner-friendly method. We start with 1 and find its “partner,” then move to 2, and so on. Connecting them makes a rainbow!

1. Start with 1: 1 x 36 = 36. So, 1 and 36 are a factor pair. Write them at opposite ends of your list.
2. Try 2: 36 ÷ 2 = 18. No remainder! So, 2 and 18 are a factor pair. Add them inside your list: 1, 2, …, 18, 36.
3. Try 3: 36 ÷ 3 = 12. No remainder! Add 3 and 12: 1, 2, 3, …, 12, 18, 36.
4. Try 4: 36 ÷ 4 = 9. No remainder! Add 4 and 9: 1, 2, 3, 4, …, 9, 12, 18, 36.
5. Try 5: 36 ÷ 5 = 7 R1. Has a remainder. So, 5 is not a factor.
6. Try 6: 36 ÷ 6 = 6. No remainder! This is a special pair where the partners are the same number.
7. You can stop! Once you reach a repeated factor (like 6), you know you’ve found all factors.

[Diagram: Rainbow arcs connecting factor pairs: 1–36, 2–18, 3–12, 4–9, and a loop on 6.]
Alt Text: A rainbow diagram showing all factor pairs of the number 36: 1 and 36, 2 and 18, 3 and 12, 4 and 9, and 6 by itself.

Your complete list from smallest to largest is: 1, 2, 3, 4, 6, 9, 12, 18, 36. Great job!

Method 2: Using Division

This method directly tests the definition. We simply divide 36 by different numbers and check for a remainder of 0.

• Is 4 a factor? 36 ÷ 4 = 9 R0. Yes!
• Is 5 a factor? 36 ÷ 5 = 7 R1. No.
• Is 9 a factor? 36 ÷ 9 = 4 R0. Yes!

You would test numbers systematically from 1 up to 36. It’s simple, but the Rainbow Method is often faster because it finds factors in pairs.

Method 3: Prime Factorization (The Building Blocks)

This method is like finding a number’s DNA! We break 36 down into its smallest prime factors (numbers only divisible by 1 and themselves, like 2, 3, 5, 7).

1. Find two factors of 36. Let’s start with 6 and 6.
2. Break down non-prime factors. 6 can be broken into 2 and 3.
3. Stop at prime numbers. Now we only have prime numbers (2, 2, 3, 3).

The prime factorization is 2 x 2 x 3 x 3, or 2² x 3².

How does this give us ALL factors? Combine the prime factors in different ways:

• 2 = 2
• 3 = 3
• 2 x 2 = 4
• 2 x 3 = 6
• 3 x 3 = 9
• 2 x 2 x 3 = 12
• 2 x 3 x 3 = 18
• 2 x 2 x 3 x 3 = 36
• Don’t forget the factor 1! (It’s the factor for every number)

You get the same list: 1, 2, 3, 4, 6, 9, 12, 18, 36.

Method 4:Factor of 36 Tree

A factor tree graphically represents prime factorization:

• Start with 36, split into 2 and 18.
• Split 18 into 2 and 9.
• Split 9 into 3 and 3.

This factor tree shows that 36 prime factors are 2 and 3, each squared.

Why Use Prime Factorization?

• Identifies prime factors efficiently.
• Helps calculate the total number of factors .

Factor Pairs of 36

Factor pairs are two numbers that multiply to the original number.

Positive Factor Pairs: all the positive factors of 36

Pair	Product
(1, 36)	36
(2, 18)	36
(3, 12)	36
(4, 9)	36
(6, 6)	36

Negative Pair Factors

Pair	Product
(-1, -36)	36
(-2, -18)	36
(-3, -12)	36
(-4, -9)	36
(-6, -6)	36

Note: Negative pair factors require two negative numbers to yield a positive product. For instance, (-2) × (-18) = 36.

Common Factors of 36

Common factors are shared between 36 and another number. These are crucial for simplifying fractions or solving equations.

Example 1: Common Factors of 24 and 36

• 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
• Greatest Common Factor (GCF): 12

Example 2: Common Factors of 18 and 36

• Factors of 18: 1, 2, 3, 6, 9, 18
• Common factors: 1, 2, 3, 6, 9, 18
• GCF: 18

Solved Examples about factors of 36.

Example 1: List all composite factors of 36.

Solution:
Composite factors have more than two factors:
4, 6, 9, 12, 18, 36.

Example 2: Verify if 15 is a factor of 36.

Calculation:
36 ÷ 15 = 2.4 → Remainder exists.
Conclusion: 15 is not a factor.

Example 3: Use prime factorization to find the GCF of 36 and 60.

Steps:

• 36 = 2² × 3²
• 60 = 2² × 3 × 5
• Common primes: 2² × 3
• GCF: 2² × 3 = 12
• math and algebra hits.

Practice factors of 36

Ready to be a Factor Finder? Try these challenges:

1. List all the factors of 24.
2. What is the prime factorization of 48?
3. How many factors does the number 18 have?

Answers:

1. Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24.
2. Prime factorization of 48: 2 x 2 x 2 x 2 x 3 = 2⁴ x 3.
3. Factors of 18: 1, 2, 3, 6, 9, 18. So, it has 6 factors.

Mastering Factors and Multiples

Understanding how to identify factors is a core competency within the Common Core State Standards for Mathematics. This skill is primarily introduced in Grade 4 (4.OA.B.4), where students learn to find all factor pairs for whole numbers in the range 1–100. It is further refined in Grade 6 (6.NS.B.4) as students apply these concepts to find the Greatest Common Factor (GCF) and solve real-world problems.

Factor Reference Table

Number	Quick Link to Factor Guide
9	Factors of 9
10	Factors of 10
21	Factors of 21
24	Factors of 24
36	Factors of 36 (this)
48	Factors of 48
60	Factors of 60

Conclusion

The factors of 36—both positive and negative—illustrate foundational mathematical principles. Through prime factorization, we uncover its prime factors (2 and 3), while factor pairs and divisibility rules provide practical tools for problem-solving.

Whether calculating the GCF, simplifying fractions, or exploring algebraic equations, understanding factors empowers you to tackle diverse challenges. Remember, the whole numbers like 36 are more than digits; they are gateways to logical thinking and analytical mastery.

Explore these related concepts:

• Factors of 60
• Factors of 9
• Factors of 48

FAQ About Factors of 36