Prime Factorization- How to Do Prime Factorization on Prime Numbers

What Prime Factorization Actually Is

Prime factorization is breaking down a number into its prime building blocks. Every integer greater than 1 is either prime or can be written as a product of primes. That's not theory—it's just how numbers work.

The process gives you the prime numbers that, when multiplied together, produce your original number. For example, 12 = 2 × 2 × 3. Those three primes are the factorization.

Prime Numbers: The Building Blocks

You need to know your primes to do this. A prime number has exactly two divisors: 1 and itself. That's it.

The primes under 20: 2, 3, 5, 7, 11, 13, 17, 19

1 is not prime. It only has one divisor. This trips people up, so remember it.

The Division Method: Step by Step

Here's how to actually do it.

Step 1: Start with the smallest prime

Try dividing by 2 first. Keep dividing by 2 until the result isn't even anymore.

Step 2: Move to the next prime

When you can't divide by 2 anymore, try 3. Then 5, then 7. Work your way up.

Step 3: Keep going until you hit 1

You're done when your result is 1.

Example: Factorize 60

60 ÷ 2 = 30
30 ÷ 2 = 15
15 ÷ 3 = 5
5 ÷ 5 = 1

Result: 60 = 2 × 2 × 3 × 5

That's it. No magic.

The Factor Tree Method

Some people prefer visual factor trees. Draw the number, branch down to two factors, keep branching until you hit primes.

For 60:

        60
       /  \
      6    10
     / \   / \
    2   3 2   5

Same result. Just visual.

Quick Comparison of Methods

Divisibility Rules That Actually Help

• 2: Last digit is even (0, 2, 4, 6, 8)
• 3: Sum of digits divisible by 3
• 5: Ends in 0 or 5
• 7: No easy rule—just try dividing
• 11: Alternating sum trick works but it's slower than just dividing

Prime Factorization of Prime Numbers

Here's the part that confuses people: a prime number is already fully factorized.

13 = 13. That's it. No decomposition needed.

7 = 7. Done.

19 = 19.

The factorization of a prime is itself. This is because primes can't be broken down further. That's literally what makes them prime.

Getting Started: Your First Practice Problems

Try these. Answers below—don't peek until you've tried.

1. Factorize 18
2. Factorize 45
3. Factorize 100
4. Factorize 17 (trick question)

Answers

18 = 2 × 3 × 3

45 = 3 × 3 × 5

100 = 2 × 2 × 5 × 5

17 = 17 (it's prime)

Common Mistakes to Avoid

• Including 1 in your factorization—don't
• Forgetting to check 2 (the only even prime) first
• Stopping before you hit 1
• Including composite numbers in your final answer

When You'll Actually Use This

Prime factorization isn't abstract busywork. It shows up in:

• Finding GCF and LCM — the fastest method uses prime factorizations
• Simplifying fractions — cancel prime factors
• Cryptography — RSA encryption relies on factoring huge numbers being hard
• Reducing square roots — √60 = √(2×2×3×5) = 2√15