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Factor Calculator
List every positive factor pair of a whole number, with prime factorization steps, a factor tree, and a prime, composite, or neither check.
A positive whole number up to 1,000,000,000
84 is composite, with 12 positive factors.
Formula
Test d = 1, 2, 3, … through floor(√n). When n ÷ d is a whole number, d and n ÷ d are a factor pair.
With your number
√84 ≈ 9.165, so floor(√84) = 9. Test d from 1 through 9.
Prime factorization
84 = 2 × 2 × 3 × 7
84 is composite. It has 12 positive factors.
Factor pairs (6)
- 1 × 84 = 84
- 2 × 42 = 84
- 3 × 28 = 84
- 4 × 21 = 84
- 6 × 14 = 84
- 7 × 12 = 84
All positive factors (12)
- 1
- 2
- 3
- 4
- 6
- 7
- 12
- 14
- 21
- 28
- 42
- 84
Exponents 2, 1, and 1 give (2 + 1) × (1 + 1) × (1 + 1) = 12 positive factors.
Trial division
Divide by the smallest prime that leaves a remainder of 0, then repeat with the quotient.
- 84 ÷ 2 = 42
- 42 ÷ 2 = 21
- 21 ÷ 3 = 7
7 is prime, so the factorization stops. 84 = 2² × 3 × 7.
Factor tree
Filled nodes are prime. Outlined nodes split into the pair drawn below them.
This factor calculator lists every positive factor of a whole number as pairs that multiply back to the original. It writes the prime factorization as powers, shows each trial-division step, draws a factor tree, and reports whether the number is prime, composite, or neither. Students use it to check work on factors and prime factorization. Whole numbers up to one billion are accepted, and a larger entry is explained on the page.
Factor pairs from trial division
A positive factor divides a whole number with nothing left over. This calculator tests the whole numbers from 1 through the greatest whole number that is at most the square root. For 84, the square root is about 9.165, so the tests run from 1 through 9. Whenever a test divides evenly, that number and the quotient form a factor pair: 1 × 84, 2 × 42, 3 × 28, 4 × 21, 6 × 14, and 7 × 12. Those pairs list every positive factor: 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, and 84.
Prime factorization, the tree, and the prime check
Prime factorization divides by the smallest prime that fits and repeats. 84 ÷ 2 = 42, 42 ÷ 2 = 21, and 21 ÷ 3 = 7, with 7 prime, so 84 = 2² × 3 × 7. The factor tree draws the same splits, starting at 84 and ending at the primes 2, 2, 3, and 7. A number is prime when its only positive factors are 1 and itself, such as 17 and 97, and composite when further factors appear, such as 84 and 360. The number 1 has a single positive factor, so the check reads neither, and there is no prime factorization to write.
Frequently Asked Questions
What is a factor pair?
A factor pair is two positive whole numbers that multiply to the number you entered. The pairs of 84 are 1 and 84, 2 and 42, 3 and 28, 4 and 21, 6 and 14, and 7 and 12. A perfect square such as 36 includes a pair made of two equal factors, 6 and 6.
How is prime factorization written?
Repeated primes are collected into a power, so the trial division 2 × 2 × 3 × 7 is written 84 = 2² × 3 × 7. The steps on the page show each division that produced those primes.
Why is 1 neither prime nor composite?
A prime number has exactly two positive factors, 1 and itself, and a composite number has more than two. The number 1 has a single positive factor, so the check reads neither, and there is no prime factorization to write.
What does the factor tree show?
The tree starts at your number and splits off the smallest prime factor, then repeats with the quotient until every branch ends at a prime. For 360 the leaves are 2, 2, 2, 3, 3, and 5, and those leaves multiply back to 360.
How large a number can I enter?
The box accepts positive whole numbers up to 1,000,000,000. Trial division only tests divisors up to the square root, which is 31,622 at that limit, so the page stays quick. A larger entry is explained in words and left uncalculated.