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 a prime number and no further division is possible.
Finding all factor pairs follows from the prime factorization. If n = 2^a x 3^b x 5^c, then 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.
For quadratics that do not factor over the integers (discriminant is not a perfect square), use the quadratic formula calculator to find the exact roots.
To find the greatest common factor of two or more numbers from their prime factorizations, see the GCF calculator.
For authoritative background on prime factorization, see Wolfram MathWorld on prime factorization.
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.
Editor at Encore Editorial, Chris Terry sets the editorial standards here and turns dense topics into plain English. He has written widely on education, finance, and consumer markets.