Prime Factorization Examples- Learn with Easy Problems

What Is Prime Factorization?

Prime factorization is breaking down a composite number into the prime numbers that multiply together to make it. That's it. No fancy definitions.

Every number bigger than 1 is either prime or composite. You need to know the difference before you can do anything useful.

Quick Refresher: Prime vs. Composite

• A prime number has exactly two divisors: 1 and itself. Examples: 2, 3, 5, 7, 11, 13.
• A composite number has more than two divisors. Examples: 4, 6, 8, 9, 12.
• The number 1 is neither prime nor composite. It stands alone.
• The number 2 is the only even prime. After that, all primes are odd.

Two Methods to Find Prime Factorization

You have two main approaches. Both work. Pick whichever makes sense to you.

Method 1: Division Method

Divide the number by prime numbers starting from 2, working your way up until you hit 1.

Steps:

1. Start with the smallest prime (2).
2. Divide the number. Write down the result.
3. Repeat with the result until you get 1.
4. List all the divisors you used.

Method 2: Factor Tree Method

Draw branches splitting numbers into factors until all branches end in primes.

Steps:

1. Write the number at the top.
2. Branch it into any two factors.
3. Keep branching composite numbers.
4. Circle or highlight the primes at the ends.
5. Read off the prime factors.

Step-by-Step Prime Factorization Examples

Example 1: Find the Prime Factorization of 12

Division Method:

• 12 ÷ 2 = 6
• 6 ÷ 2 = 3
• 3 ÷ 3 = 1

Prime factors: 2 × 2 × 3

Or written with exponents: 2² × 3

Factor Tree Method:

12
/ \
3 4
/ \
2 2

Same result: 2 × 2 × 3

Example 2: Find the Prime Factorization of 36

Division Method:

• 36 ÷ 2 = 18
• 18 ÷ 2 = 9
• 9 ÷ 3 = 3
• 3 ÷ 3 = 1

Prime factors: 2 × 2 × 3 × 3

With exponents: 2² × 3²

Example 3: Find the Prime Factorization of 45

Division Method:

• 45 ÷ 3 = 15
• 15 ÷ 3 = 5
• 5 ÷ 5 = 1

Prime factors: 3 × 3 × 5

With exponents: 3² × 5

Notice 45 is odd, so you skip 2 entirely and start at 3.

Example 4: Find the Prime Factorization of 72

Division Method:

• 72 ÷ 2 = 36
• 36 ÷ 2 = 18
• 18 ÷ 2 = 9
• 9 ÷ 3 = 3
• 3 ÷ 3 = 1

Prime factors: 2 × 2 × 2 × 3 × 3

With exponents: 2³ × 3²

Example 5: Find the Prime Factorization of 100

Division Method:

• 100 ÷ 2 = 50
• 50 ÷ 2 = 25
• 25 ÷ 5 = 5
• 5 ÷ 5 = 1

Prime factors: 2 × 2 × 5 × 5

With exponents: 2² × 5²

This one also equals 10² since 10 = 2 × 5. Useful to remember for square root problems.

Example 6: Find the Prime Factorization of 126

Division Method:

• 126 ÷ 2 = 63
• 63 ÷ 3 = 21
• 21 ÷ 3 = 7
• 7 ÷ 7 = 1

Prime factors: 2 × 3 × 3 × 7

With exponents: 2 × 3² × 7

Quick Reference Table

How to Check Your Work

Multiply all the prime factors back together. If you get the original number, you're right.

Example: For 36, you got 2² × 3².

Check: 2 × 2 × 3 × 3 = 4 × 9 = 36 ✓

That's it. No other way to verify.

Common Mistakes to Avoid

• Including 1 — It's not a prime. Stop writing "1" in your factor list.
• Forgetting to finish — Keep dividing until you hit 1. Many people stop at a composite number.
• Using composite factors — If you use 4 as a factor, break it down further. 4 = 2 × 2.
• Skipping primes — If 2 doesn't divide evenly, try 3. Then 5. Then 7. Don't jump ahead.

Where Prime Factorization Shows Up

You need this skill for:

• Finding GCF (Greatest Common Factor) — Identify common prime factors between numbers
• Finding LCM (Least Common Multiple) — Use prime factors to find the smallest shared multiple
• Simplifying fractions — Cancel common prime factors
• Square roots — Pull out pairs of identical primes (√36 = 2 × 3 = 6)

Practice Problems

Try these on your own before checking:

1. Prime factorization of 48
2. Prime factorization of 75
3. Prime factorization of 84
4. Prime factorization of 144

Answers

48: 2 × 2 × 2 × 2 × 3 = 2⁴ × 3

75: 3 × 5 × 5 = 3 × 5²

84: 2 × 2 × 3 × 7 = 2² × 3 × 7

144: 2 × 2 × 2 × 2 × 3 × 3 = 2⁴ × 3² (which is also 12²)