Factors, Multiples & Primes
Prime Factorization – Breaking Numbers into Prime Pieces
Prime factorization is the process of writing a number as a product of prime numbers. Because primes cannot be broken down further, this gives us the deepest possible view of a number's structure.
Finding the prime factorization of a small number like 84 takes seconds by hand. But the difficulty grows explosively with size: factoring a 10-digit number by computer is instant, a 100-digit number takes noticeably longer, and a carefully chosen 300-digit number can keep even powerful computers busy for far longer than is practical using known methods. That asymmetry — multiplying primes together is easy, but undoing the multiplication is brutally hard — is the entire foundation of RSA encryption, the system that secures much of your online banking and messaging today.
What Is It?
Prime factorization of N = the unique expression N = p¹ × p² × p³ × ... where every p is a prime number.
Step by Step: Finding a Prime Factorization by Division
The quickest hand method is repeated division: keep dividing by the smallest prime that fits, and stop only when what’s left over is 1.
- Start with your number. Find the smallest prime that divides it exactly — always test 2 first, then 3, then 5, then 7, and so on, in order.
- Divide the number by that prime. Write the prime down as one factor, and carry the answer forward as your new working number.
- Repeat step 1 on the new, smaller number. The smallest prime that works might be the same one again, or it might be the next one up.
- Keep going until the working number reaches 1. Every prime you wrote down along the way, multiplied together, is the prime factorization.
- Check your work: multiply all the primes you found back together — you should land exactly back on the number you started with.
| Prime | How to tell it divides evenly |
|---|---|
| 2 | The number is even (ends in 0, 2, 4, 6 or 8). |
| 3 | The digits add up to a multiple of 3. |
| 5 | The number ends in 0 or 5. |
| 7, 11, 13… | No simple shortcut — just divide and see if it comes out exact. |
Worked Examples
36 ÷ 2 = 18 → 18 ÷ 2 = 9 → 9 ÷ 3 = 3 → 3 is prime.
36 = 2 × 2 × 3 × 3 = 2² × 3²
84 ÷ 2 = 42 → 42 ÷ 2 = 21 → 21 ÷ 3 = 7 → 7 is prime.
84 = 2² × 3 × 7
360 ÷ 2 = 180 → ÷2 = 90 → ÷2 = 45 → 45 ÷ 3 = 15 → 15 ÷ 3 = 5 → 5 prime.
360 = 2³ × 3² × 5
More Examples, Laid Out as a Division Ladder
The exact same repeated-division process can be written as a ladder — the smallest prime on the left, the shrinking number on the right, one rung per division. This is especially useful once a number needs four or five divisions to fully break down.
| Divide by | 420 |
|---|---|
| 2 | 210 |
| 2 | 105 |
| 3 | 35 |
| 5 | 7 |
7 is already prime — stop here.
Check: 4 × 3 × 5 × 7 = 12 × 35 = 420 ✓
128 ÷ 2 = 64 → ÷2 = 32 → ÷2 = 16 → ÷2 = 8 → ÷2 = 4 → ÷2 = 2 → ÷2 = 1.
Every single division used the prime 2 — seven times in a row.
Not every number splits into a mix of different primes — some, like 128, are built from just one prime repeated many times over. That’s still a perfectly valid (and unique) prime factorization.
Test 2: 97 is odd, doesn’t divide. Test 3: digits sum to 16, not a multiple of 3. Test 5: doesn’t end in 0 or 5. Test 7: 97÷7 ≈ 13.86, not exact. Since √97 ≈ 9.8, there’s no need to test any prime larger than 9 — if none of 2, 3, 5, 7 divide it, nothing bigger will either.
Every prime number is its own prime factorization — there’s nothing left to break down.
Using Prime Factorization for GCF and LCM
| Goal | Rule | Example (12=2²×3, 18=2×3²) |
|---|---|---|
| GCF | Take lowest power of each shared prime | 2¹ × 3¹ = 6 |
| LCM | Take highest power of every prime | 2² × 3² = 36 |
Both of these techniques — along with the ladder (division) method and several more worked examples — are covered in full depth on the dedicated Methods for Finding the GCF and Methods for Finding the LCM pages.
The Uniqueness Guarantee
The Fundamental Theorem of Arithmetic guarantees that every whole number greater than 1 has exactly one prime factorization (order of primes does not count). This uniqueness makes it a reliable tool — it’s also exactly why every worked example on this page has just one correct answer, no matter which order you happen to test the primes in along the way.
Key Takeaways
- Divide repeatedly by the smallest prime until 1 is reached.
- Write the result using exponents for repeated primes.
- A number made of just one repeated prime (like 128 = 2⁷) still has a valid, unique prime factorization.
- A prime number’s prime factorization is simply itself.
- Every whole number > 1 has a unique prime factorization.
- Use lowest powers for GCF; highest powers for LCM.