This prime factorization calculator breaks a whole number into a product of primes. The default n = 840 returns 2 × 2 × 2 × 3 × 5 × 7. That product multiplies back to 840 and lists every prime with the right count.
Use it for homework checks, factor trees, and prep work before GCF or LCM. For all positive divisors rather than primes only, open the factor calculator. For greatest common factor steps, use the GCF calculator.
How the formula works
Divide by 2 while even, then try the next integers up to the square root, accepting only prime hits, until the remaining cofactor is 1 or prime. The collected primes are the factorization.
Worked example
840 ÷ 2 = 420, ÷ 2 = 210, ÷ 2 = 105, ÷ 3 = 35, ÷ 5 = 7, and 7 is prime. The full product is 2 × 2 × 2 × 3 × 5 × 7.
| Step | Result |
|---|---|
| Start | 840 |
| Divide by 2 three times | 105 |
| Divide by 3 | 35 |
| Divide by 5 | 7 |
| Prime factorization | 2 × 2 × 2 × 3 × 5 × 7 |
How to use the fields
- Enter a positive whole number as n.
- Read the prime product with repeated factors shown.
- Multiply the primes back to verify the original n.
Why multiplicity matters
Writing 840 = 2 × 3 × 5 × 7 would be incomplete because three factors of 2 are required. Exponent form 2³ × 3 × 5 × 7 is the compact twin of the expanded list.
Common mistakes
- Stopping after the first prime factor and forgetting residual composites
- Treating 1 as a prime factor
- Dropping repeated primes when copying a factor tree
- Confusing prime factorization with a full divisor list
Link to GCF and LCM
Once two numbers are factored, GCF keeps the lowest shared powers and LCM keeps the highest powers across primes. Factorization is the shared language for both.
Classroom factor trees
Different trees can branch in different orders and still finish with the same primes. Ask students to compare trees for 840 and confirm the multiset of primes matches.
Checking by multiplication
Always multiply the listed primes as a final audit. If the product is not n, a factor was dropped or a composite slipped into the list.
Larger composites
Harder composites take more division steps, but the method is the same. Keep primes ordered small to large so classmates can compare answers quickly.
When a remaining cofactor is itself prime and larger than the last trial divisor, append it and stop.
Exponent form vs expanded form
Teachers may accept 2³ × 3 × 5 × 7 or the expanded 2 × 2 × 2 × 3 × 5 × 7. Both encode the same multiset of primes. Expanded form makes multiplication checks obvious; exponent form is shorter on a test sheet.
When grading, look for matching exponents on each prime rather than matching only the final product, so a lucky wrong factorization that still multiplies to 840 does not slip through.
Square free vs repeated primes
A square free integer has no repeated prime factors. 840 is not square free because 2 appears three times. Spotting squares and higher powers early can speed mental factoring for contest problems.
Word problems that hide composites
Area tiles, gear teeth counts, and package sizes often ask you to factor first so later GCF or LCM steps are clean. Run the factorization, label each prime, then continue the story problem without reintroducing the original composite until the end.
If two quantities share a factorization step, write both factor lists side by side before computing shared or highest powers.
Divisibility warmups
Before long division, test evenness, digit sums for 3, and ending digits for 5. Those filters recreate the 840 path quickly: three divisions by 2, one by 3, one by 5, and a leftover 7.
Students who skip divisibility tests often jump to larger trial divisors and miss repeated small primes, which is exactly how incomplete products appear.
Connecting to percent checks
After you factor, you might still need to compare sizes or shares. A separate percentage calculator handles percent language; keep prime factorization for the integer structure underneath.
Limitations
The tool targets ordinary whole number homework scale factorization. It is not a cryptography grade factoring service, and it does not rewrite decimals or fractions.