How it works
Prime factorization breaks a number into the prime numbers that multiply together to make it. The calculator uses trial division: it tests each prime (2, 3, 5, 7, 11, …) to see if it divides the number evenly, records how many times it divides, and repeats with the quotient until nothing remains but 1.
For small to moderately large numbers, this method is fast and straightforward. The result shows both the prime factors and their exponents (how many times each prime appears).
The formula
n = p₁^a₁ × p₂^a₂ × p₃^a₃ × … × pₖ^aₖ
where p₁, p₂, p₃ … pₖ are distinct prime numbers and a₁, a₂, a₃ … aₖ are their exponents (powers).
Worked example
Let's factorize 360.
Step 1: Is 360 divisible by 2?
Yes. 360 ÷ 2 = 180. Record one factor of 2.
Step 2: Is 180 divisible by 2?
Yes. 180 ÷ 2 = 90. Record another factor of 2.
Step 3: Is 90 divisible by 2?
Yes. 90 ÷ 2 = 45. Record another factor of 2.
Step 4: Is 45 divisible by 2?
No. Move to the next prime, 3.
Step 5: Is 45 divisible by 3?
Yes. 45 ÷ 3 = 15. Record one factor of 3.
Step 6: Is 15 divisible by 3?
Yes. 15 ÷ 3 = 5. Record another factor of 3.
Step 7: Is 5 divisible by 3?
No. Move to 5.
Step 8: Is 5 divisible by 5?
Yes. 5 ÷ 5 = 1. Record one factor of 5.
Result: 360 = 2³ × 3² × 5
Or written as a list: 2, 2, 2, 3, 3, 5
You can verify: 8 × 9 × 5 = 360. ✓
Common mistakes
Forgetting to repeat: Many people divide by a prime once and move on. Always keep dividing by the same prime until it no longer divides evenly.
Including 1 in the answer: Since 1 is not prime, it should never appear in a prime factorization. If your result includes 1, you've made an error.
Stopping early: If your final quotient is greater than 1 and isn't prime, keep factoring. For instance, if you get down to 6, that's not prime—it factors as 2 × 3.
Confusing prime and composite: A prime has exactly two divisors (1 and itself). A composite has more. The number 1 is neither, and 2 is the only even prime.