What the factors of a number are
A factor of a whole number is a whole number that divides it exactly, with no remainder. This is the integer task: finding the numbers that go into 15, 36 or 120. It is different from factoring a polynomial such as x² + 5x + 6, which is algebra and needs a different tool.
Take 15. It divides by 1, 3, 5, 15, so 15 has 4 factors. Check each one: 15 ÷ 3 = 5 and 15 ÷ 5 = 3, but 15 ÷ 2 leaves a remainder. The calculator tests every divisor up to the square root and pairs each one with its partner, so even a number with twelve digits is handled instantly. The number 120 has 16 factors: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120. Their sum is 360.
Factor pairs and perfect squares
A factor pair is two whole numbers whose product is the original number. The factors of 120 arrange themselves into 8 pairs: 1 × 120, 2 × 60, 3 × 40, 4 × 30, 5 × 24, 6 × 20, 8 × 15, 10 × 12. Each factor appears in exactly one pair, which is why the factors come in matched twos.
The exception is a perfect square. For 36 the pairs are 1 × 36, 2 × 18, 3 × 12, 4 × 9, 6 × 6, and the middle pair is 6 × 6, where a factor is paired with itself. That is why a perfect square always has an odd number of factors (9 for 36) while other numbers have an even number. The square root counts once, not twice.
Prime factorization and the factor tree
A prime number has exactly two factors, 1 and itself. 97 is prime: its only factors are 1, 97. Every other whole number above 1 can be written as a product of primes in exactly one way, ignoring order. That is the prime factorization.
To build it, divide by the smallest prime that fits, then repeat on what is left. For 120: 120 ÷ 2 = 60, 60 ÷ 2 = 30, 30 ÷ 2 = 15, 15 ÷ 3 = 5, and 5 is prime. So 120 = 2³ × 3 × 5. The factor tree shows these splits as branches, with each prime at the end of a branch. Exponents count repeats: 100 = 2² × 5².
The prime factorization also predicts the number of factors. Add 1 to each exponent and multiply: for 120 = 2³ × 3 × 5 that is (3 + 1)(1 + 1)(1 + 1) = 16, matching the list above.
Negatives, zero and the greatest common factor
For a negative number the factors are the same as for its positive twin, and each can carry either sign. The prime form of −12 is -1 × 2² × 3, with −1 written out so the sign is not lost. The number 1 has just one factor, itself (1 factor). Zero is excluded because every nonzero number divides it, so its list never ends.
Do not confuse this page with the greatest common factor. That compares two or more numbers and returns the largest factor they share; here you list everything that divides a single number. When you need the shared one, use the GCF calculator, or the LCM calculator for common multiples.
Why factors matter
Factors are the building blocks behind several everyday jobs. Simplifying a fraction means dividing the top and bottom by a shared factor, and finding a common denominator relies on the multiples of those same numbers. Factoring also helps with divisibility tests: if you know a number is divisible by both 3 and 4, it is divisible by 12. In number theory, the fact that every whole number above 1 has exactly one prime factorization is what makes primes so important. A number with only two factors, 1 and itself, is prime, and a number with more is composite. Checking small primes in order, as the calculator does, is the simplest reliable way to see which case you have.
Frequently asked questions
Is 1 a prime number?
No. A prime needs exactly two different factors, and 1 has only one. That also keeps the prime factorization of every number unique.
How many factors does zero have?
Infinitely many nonzero numbers divide 0, so it has no finite list and the calculator does not accept it.
Does a square root count twice?
No. In 6 × 6 the factor 6 is counted once in the list of factors, though it fills both places in its pair.
How is this different from the GCF?
The GCF finds the largest factor common to several numbers. This tool lists every factor of one number.
If you are dividing numbers by hand, the long division calculator shows the quotient and remainder, and all calculators lists the rest of our maths tools.