Enter any positive integer to get its prime factorization and all factor pairs. Or enter a quadratic (ax^2 + bx + c) to factor it into two binomials. Every step is shown.
Try 360 or 1001. Negative numbers use the absolute value.
To factor a number means to write it as a product of smaller integers. The prime factorization is the most reduced version: every factor is prime and no further division is possible. Factoring a quadratic runs the same idea in reverse, rebuilding a trinomial from two binomials that multiply back into it.
Finding all factor pairs follows from the prime factorization. If n = 2^a x 3^b x 5^c, the total count of factors is (a+1)(b+1)(c+1). Each factor pair (d, n/d) is found by testing every divisor from 1 up to the square root of n, which is why the search stops early instead of checking all the way to n.
With integer factoring, people often stop trial division too early and mislabel a composite number as prime, so always check up to the square root of what remains before calling it done. With quadratics, a frequent error is picking two numbers that add correctly but multiply to the wrong product, or the reverse; both conditions have to hold at once. If no integer pair satisfies both, the quadratic likely does not factor over the integers, and the quadratic formula calculator will find the exact roots instead.
To find the greatest common factor of two or more numbers using the same prime factorizations, see the GCF and LCM calculator.
For authoritative background on prime factorization, see Wolfram MathWorld's page on prime factorization.
Round a messy result, or check whether a number simplifies under a radical.
Prime factorization breaks a number into a product of prime numbers. For example, 60 = 2 x 2 x 3 x 5, usually written as 2^2 x 3 x 5. Every integer above 1 has exactly one prime factorization (the Fundamental Theorem of Arithmetic).
Factor pairs are pairs of integers that multiply together to give the original number. For 36, the pairs are (1, 36), (2, 18), (3, 12), (4, 9), and (6, 6). To find them, test every integer from 1 up to the square root of the number.
For ax^2 + bx + c, find two numbers that multiply to (a x c) and add to b. Use those to split the middle term, then factor by grouping. For x^2 + 5x + 6, find numbers that multiply to 6 and add to 5: those are 2 and 3, so the factored form is (x + 2)(x + 3).
A quadratic ax^2 + bx + c does not factor over the integers when its discriminant (b^2 - 4ac) is not a perfect square. In that case, use the quadratic formula calculator to find its roots.
Factors divide evenly into a number with no remainder. Multiples are what you get when you multiply a number by positive integers. The factors of 12 are 1, 2, 3, 4, 6, and 12. The multiples of 12 are 12, 24, 36, 48, and so on.
Marcus Vance spent a decade building data pipelines and testing calculation logic before he started writing about the math behind the tools. On MathCalcTools he checks every formula against the source math, then writes out the steps so a reader can follow the same path a calculator takes.