The formula this page uses
n = p₁^a₁ × p₂^a₂ × … × pₖ^aₖ, with every pᵢ prime and every aᵢ a whole number above 0
Prime Factorization Calculator
Use the free prime factorization calculator to work through the number relationship, preserve exact values, and check the result with estimation. Choose a focused operation below to calculate, inspect the method, and connect the result to a visual model.
The visual updates with your calculation.
Result—Choose a mode, enter the known values, and calculate.
Show your work Step-by-step method
- The calculation method and verification will appear here.
Arithmetic
Prime Factorization Calculator explained
The short version
- Every whole number above 1 breaks into primes in exactly one way, no matter how you start splitting it.
- A prime is a number with no factors except 1 and itself: 2, 3, 5, 7, 11, 13 and so on.
- You only ever have to test primes up to the square root of the number.
What each part means
SymbolWhat it means
n — The numberAny whole number above 1. The number 1 has no prime factorisation at all.
p — A prime factorOne of the building blocks. Written smallest first by habit, not by rule.
a — The powerHow many times that prime appears. 2³ means the 2 was used three times.
√n — The stopping pointOnce your test prime passes the square root of n, you can stop looking.
Show your work: a full example
- Break down 360360
- Divide by 2360 ÷ 2 = 180
- Divide by 2 again180 ÷ 2 = 90
- And again90 ÷ 2 = 45
- 45 is odd, so move to 345 ÷ 3 = 15
- Divide by 3 again15 ÷ 3 = 5
- 5 is prime, so stop360 = 2 × 2 × 2 × 3 × 3 × 5 = 2³ × 3² × 5
A second, different case
- A different case: the number turns out to be primeis 97 prime?
- Find the stopping point√97 = 9.85, so only test primes up to 9
- Test 297 is odd, so 2 is out
- Test 397 ÷ 3 = 32 remainder 1
- Test 597 ÷ 5 = 19 remainder 2
- Test 797 ÷ 7 = 13 remainder 6
- Nothing divides it97 = 97, so 97 is prime and its factorisation is just itself
Copy-ready example
360
Break down 360
Questions about the Prime Factorization Calculator
Does the order I split the number in change the answer?
No, and that is the surprising part. Starting 360 as 36 × 10 or as 8 × 45 or as 2 × 180 all end at 2³ × 3² × 5. This is called unique factorisation, and it is the reason primes are treated as the building blocks of arithmetic.
Why is 1 not counted as prime?
Because it would break the one-answer rule. If 1 were prime you could write 12 as 2² × 3, or 1 × 2² × 3, or 1 × 1 × 2² × 3, forever. Leaving 1 out keeps every number with exactly one factorisation.
How do I get every factor from the prime factorisation?
Add one to each power and multiply those together. For 360 = 2³ × 3² × 5, that is 4 × 3 × 2 = 24 factors. The 4 comes from being able to use zero, one, two or three 2s, and the same idea covers the other primes.
What is the biggest prime I need to test?
The largest prime that is not bigger than the square root. For a number near 400 that means testing up to 19, because 20² = 400. That is only eight primes to check, which is why the square-root rule saves so much time.
Calculate, interpret, verify.
This workspace keeps the formula and the meaning together. Decimal results are rounded for display; retain full precision when you continue a calculation.
Enter known values
Match each input to the quantities in the problem and keep units consistent.
Use the relationship
The result panel identifies the formula or algorithm and shows the main substitutions.
Read the visual
Use the diagram, plot, or data display to check scale, direction, and plausibility.
Practice unaided
Move to targeted questions once you can explain why the method applies.
Learn the mathematics
Understand arithmetic behind this calculator
A calculator confirms an answer. Working the method yourself is what makes the next problem faster.