Prime factorization means writing a number as a product of prime numbers. For example, 60 = 2 × 2 × 3 × 5, or 2² × 3 × 5. This guide shows two reliable methods, worked examples, common mistakes, and a 10-question practice worksheet with answers. You can copy it into a printable sheet for study or classroom use.
What Is Prime Factorization?
A prime number has exactly two factors: 1 and itself (2, 3, 5, 7, 11…). A composite number has more than two factors (4, 6, 12, 60…). Prime factorization breaks a composite number down until only primes remain.
The Fundamental Theorem of Arithmetic says every whole number greater than 1 is either prime or can be written as a product of primes in exactly one way, apart from the order of the factors. So 60 can only be 2² × 3 × 5, however you find it. This is why the answer is reliable and why it is useful for other problems.
Two special cases:
- 1 is not prime (it has only one factor).
- 2 is the only even prime number.
Prime Numbers You Should Know
Memorize the primes below 50, as you will use them constantly:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47
Divisibility Rules That Speed Things Up
Test small primes in order, starting with 2.
| Prime | Rule | Example |
| 2 | Last digit is even | 84 → ends in 4 ✔ |
| 3 | Digit sum is divisible by 3 | 252 → 2+5+2 = 9 ✔ |
| 5 | Last digit is 0 or 5 | 135 ✔ |
| 7 | No quick rule for beginners. Try dividing. | 91 ÷ 7 = 13 |
Method 1: Factor Tree
A factor tree splits a number into two factors, then splits any composite factor again. You stop when every “branch end” is prime.
Steps
- Write the number at the top.
- Split it into any two factors (not 1 and itself).
- If a factor is prime, circle it. If not, split it again.
- Continue until all end values are prime.
- Multiply the circled primes to write the answer.
Example: factorize 180
- 180 = 18 × 10
- 18 = 2 × 9, and 9 = 3 × 3
- 10 = 2 × 5
The primes at the ends are 2, 3, 3, 2, 5. In order: 180 = 2 × 2 × 3 × 3 × 5 = 2² × 3² × 5.
Check: 4 × 9 × 5 = 180 ✔
A different starting split, such as 180 = 20 × 9 or 180 = 2 × 90, gives the same primes. That is the theorem at work.
Method 2: Repeated Division (Ladder Method)
Divide the number by the smallest prime that fits, then repeat with the quotient until you reach 1.
Example: factorize 84
| Divide by | Result |
| 2 | 84 ÷ 2 = 42 |
| 2 | 42 ÷ 2 = 21 |
| 3 | 21 ÷ 3 = 7 |
| 7 | 7 ÷ 7 = 1 |
Primes used: 2, 2, 3, 7. So 84 = 2² × 3 × 7.
Check: 4 × 3 × 7 = 84 ✔
Which method should you use? Factor trees are easier to see and good for beginners. Repeated division is more organized for large numbers because you never lose track of which factors are still unsplit.
Worked Examples
Example 1: 36
36 = 6 × 6 = (2 × 3) × (2 × 3) = 2 × 2 × 3 × 3 = 2² × 3²
Example 2: 210
Division: 210 ÷ 2 = 105; 105 ÷ 3 = 35; 35 ÷ 5 = 7; 7 ÷ 7 = 1.
So 210 = 2 × 3 × 5 × 7. Each prime appears once.
Check: 2 × 3 = 6, 6 × 5 = 30, 30 × 7 = 210 ✔
Example 3: Is 97 prime?
You only need to test primes whose square does not exceed 97. Since 10² = 100 is already larger than 97, test 2, 3, 5 and 7.
- 2: 97 is odd ✘
- 3: digit sum 9 + 7 = 16, not divisible by 3 ✘
- 5: ends in 7 ✘
- 7: 7 × 13 = 91 and 7 × 14 = 98, so 97 is not a multiple of 7 ✘
None divide 97, so 97 is prime.
Example 4: 1001
1001 ÷ 7 = 143; 143 ÷ 11 = 13; 13 is prime.
So 1001 = 7 × 11 × 13.
Check: 7 × 11 = 77, 77 × 13 = 1001 ✔
Writing Answers with Exponents
When a prime repeats, count how many times it appears and write that count as an exponent.
2 × 2 × 2 × 3 × 3 = 2³ × 3²
Always list primes from smallest to largest. This is the standard form, and it makes answers easy to compare and mark.
Why Prime Factorization Matters: GCF and LCM
Prime factorization gives a dependable way to find the GCF (greatest common factor, also called HCF) and the LCM (least common multiple).
Example: find the GCF and LCM of 36 and 84
- 36 = 2² × 3²
- 84 = 2² × 3 × 7
GCF: take each shared prime with the smaller exponent: 2² × 3 = 12.
LCM: take every prime that appears, with the larger exponent: 2² × 3² × 7 = 252.
Check: GCF × LCM = 12 × 252 = 3,024, and 36 × 84 = 3,024 ✔
This also helps with simplifying fractions. For example, 36/84: cancel the common factor 12 to get 3/7.
Common Mistakes to Avoid
-
Stopping too early
Mistake: Writing 36 = 4 × 9 and calling it done.
Why it happens: 4 and 9 are factors, but they are not prime.
Fix: Keep splitting: 4 = 2 × 2 and 9 = 3 × 3, so 36 = 2² × 3². -
Treating 1 as a prime
Mistake: Writing 12 = 1 × 2 × 2 × 3.
Fix: 1 is not prime. Leave it out: 12 = 2² × 3. -
Calling 9 or 15 prime because they are odd
Why it happens: Students remember that only 2 is an even prime and assume odd numbers are prime.
Fix: Odd numbers can be composite: 9 = 3 × 3 and 15 = 3 × 5. -
Adding exponents wrongly
Mistake: Writing 2 × 2 × 2 as 2 × 3 or 6.
Fix: 2 × 2 × 2 = 2³. The exponent counts how many times the prime is multiplied. -
Not checking the answer
Multiply your primes back together. If you do not get the original number, find the error.
Prime Factorization Practice Worksheet (with Answers)
Part A: Factor trees or division. Write each number as a product of primes using exponents.
- 24
- 45
- 60
- 72
- 100
Part B: Challenge.
- 126
- 150
- 225
- 252
- 360
Part C (bonus). Find the GCF and LCM of 24 and 60.
Answer Key
| # | Answer | Working summary |
| 1 | 2³ × 3 | 24 → 2, 2, 2, 3 |
| 2 | 3² × 5 | 45 = 9 × 5 |
| 3 | 2² × 3 × 5 | 60 = 4 × 15 |
| 4 | 2³ × 3² | 72 = 8 × 9 |
| 5 | 2² × 5² | 100 = 10 × 10 |
| 6 | 2 × 3² × 7 | 126 = 2 × 63 |
| 7 | 2 × 3 × 5² | 150 = 2 × 75 |
| 8 | 3² × 5² | 225 = 15 × 15 |
| 9 | 2² × 3² × 7 | 252 = 4 × 63 |
| 10 | 2³ × 3² × 5 | 360 = 8 × 45 |
Bonus: 24 = 2³ × 3 and 60 = 2² × 3 × 5. GCF = 2² × 3 = 12. LCM = 2³ × 3 × 5 = 120. Check: 12 × 120 = 1,440 = 24 × 60 ✔
How to Use a Worksheet Effectively
- Attempt first, then check. Try all questions before looking at the answers.
- Verify by multiplying. Multiply your primes back to confirm.
- Mark your errors. Note whether a mistake came from divisibility, stopping early or exponents.
- Mix methods. Do some by factor tree and some by division, then compare.
- Build up. Start with numbers under 100, then move to numbers like 252 and 360.
Key Takeaways
- Prime factorization writes a number as a product of primes, and the result is unique.
- Factor trees and repeated division both work. Use whichever you can follow reliably.
- 1 is not prime, and 2 is the only even prime.
- Write answers in ascending order with exponents, such as 2³ × 3².
- Always check by multiplying back.
- Prime factorization is the base for finding GCF, LCM and simplifying fractions.
Conclusion
Prime factorization is a skill built on a few simple habits: test small primes in order, keep splitting until only primes remain, and check your product. With regular practice using worksheets like the one above, the process becomes quick and automatic, and it prepares you for GCF, LCM and fraction work. Explore more mathematics practice resources from Digital Libraries and try the worksheet before checking the answer key.
FAQ
- What is the easiest way to do prime factorization?
For small numbers, a factor tree is the easiest to visualize. For larger numbers, repeated division by 2, 3, 5, 7 and so on keeps the work organized. - Can two students get different factor trees for the same number?
Yes, the branches may look different, but the final primes are always the same, apart from order. - Is 1 a prime number?
No. A prime needs exactly two factors, and 1 has only one. - How do I know when to stop dividing?
Stop when the remaining number is 1 (in division) or when every branch end is a prime (in a factor tree). - How do I know if a large number is prime?
Test division by primes up to the square root of the number. If none divide it evenly, it is prime. Example 3 above shows this for 97.