Prime factorization writes an integer at least 2 as a product of primes only. Continue splitting any composite factor, then multiply the final factors back together. This guide uses 360 to connect the process with divisor counts and fraction simplification.
Prime factors, distinct factors and divisors are different counts
| Question | Output | Example for 360 |
|---|---|---|
| Prime factorization | Prime product with repetitions | 2×2×2×3×3×5 |
| Distinct prime factors | Different primes appearing | 2, 3 and 5: three types |
| Exponent form | Count repeated factors | 2³×3²×5 |
| Number of positive divisors | Multiply one more than each exponent | (3+1)(2+1)(1+1)=24 |
| Primality | Exactly two positive divisors? | 360 is not prime |
1 has only one positive divisor, so it is not prime. 2 is prime; its factorization is simply 2 and its positive divisors are 1 and 2. Do not confuse distinct-prime count with total divisor count.
Step-by-step factorization of 360
- Divide 360 by 2 to get 180, then by 2 to get 90, then by 2 to get 45. There are three factors of 2.
- Divide 45 by 3 to get 15, then by 3 to get 5.
- Keep the remaining prime 5. A division ladder written through quotient 1 ends with 5/5=1.
- Collect 2×2×2×3×3×5=2³×3²×5.
- Multiply back: 8×9×5=360. Do not add 1 as a prime factor.
Check the conditions, then verify
Compare the same integer with your hand calculation
Enter 360 and press Factorize. Check the product, exponent form and 24 positive divisors independently.
Any starting factor pair works if every branch ends in primes. Starting with 360=36×10 is valid, but leaving 36 or 10 as final “prime” factors is not. OpenStax explains the factor-tree method and uniqueness apart from ordering.
Why add one to each exponent for divisor counts?
For 360=2³×3²×5¹, choose the exponent of 2 from 0–3 (four choices), 3 from 0–2 (three), and 5 from 0–1 (two). These independent choices give 4×3×2=24 positive divisors. Exponent zero includes choosing none of that prime.
| Integer | Prime factorization | Positive divisor count |
|---|---|---|
| 2 | 2¹ | 2 |
| 12 | 2²×3 | 3×2=6 |
| 180 | 2²×3²×5 | 3×3×2=18 |
| 360 | 2³×3²×5 | 4×3×2=24 |
| 1024 | 2¹⁰ | 11 |
These are positive divisors. Check the task’s convention if negative divisors are included. Review repeated-factor notation in the exponent guide.
Choose factorization, GCD, LCM or a sum
With 24=2³×3 and 36=2²×3², the greatest common divisor uses the smaller shared exponents: 2²×3=12. The least common multiple takes the larger exponent for every appearing prime: 2³×3²=72. A prime absent from one number has exponent zero there.
| Task | Next step |
|---|---|
| Split 360 into primes | This Prime Factorization tool |
| Find GCD/LCM of 24 and 36 | GCD & LCM Calculator |
| Reduce 24/36 | Fraction Calculator |
| Add 1+2+⋯+100 | A sum rather than a product: sequence guide |
Tool workflow and input limits
- Open Prime Factorization and enter one integer. The initial 360 placeholder is not an entered value.
- Press Factorize or Enter while focused in the field. Typing alone does not calculate.
- Check product form, exponent form, divisor count and primality. Multiply back to verify.
- Before copying, match the displayed result to the source integer. Editing the input clears the previous output and requires another run.
| Condition | Current support |
|---|---|
| Integer input | 2–999,999,999,999,999, at most 15 digits |
| One-digit primes such as 2 | Supported; “at least 2” is a value, not a digit count |
| 0, 1, negatives or decimals | Not accepted; 1 has no prime factors |
| Grouping commas | Removed, so do not paste 2,3 as a list of two numbers |
| Divisor list or divisor sum | Not output; the tool reports divisor count |
The 15-digit limit is not a speed guarantee. Trial-division work also depends on the size of prime factors; large primes and products of large prime factors can require more work. This is not a tool for factoring cryptographic-scale integers. Start with the small integers used in your lesson.
Check the conditions, then verify
Verify the product and count before recording the answer
Keep the original integer, prime exponents and divisor count together. Favorite the tool for another check later.
Update and verification scope
On October 27, 2026 the Tools Hub editorial team compared primary teaching material with the public source’s definitions, input limits and divisor formula. The examples 2, 12, 180, 360 and 1024 were checked by hand and browser interaction. Follow the notation required by your course and multiply back before transcribing. Favorites retain a shortcut in the same browser; they do not save or sync coursework or inputs.
Find related simplification and multiple lessons in the mathematics category.
FAQ
Can I factor 1 or 2?
1 has no prime factors and this tool starts at 2. The prime 2 factors as itself and has two positive divisors, 1 and 2.
Why does 360 have three distinct primes but 24 divisors?
The distinct primes are 2, 3 and 5. Positive divisors choose the exponent of 2 from 0–3, 3 from 0–2 and 5 from 0–1, giving 4×3×2=24 combinations.
Does the starting factor pair change the answer?
After every factor has been split into primes, the product is the same apart from order. You may start at 360=36×10, but must not leave composite 36 or 10 as final prime factors.
Is every 15-digit input fast?
There is no speed guarantee. Inputs are limited to 999,999,999,999,999, but trial-division work also depends on factor sizes. Editing clears the previous result; press Factorize or Enter to run again.
