Factors Calculator lists the positive divisors and factor pairs of a nonzero integer, including signed pairs for negative inputs.
Read the divisor list and factor pairs
A divisor divides an integer exactly with no remainder. This tool lists positive divisors of the entered integer’s absolute value, then shows factor pairs that multiply to the original signed input. Prime powers provide the decomposition behind the list. Each positive divisor comes from a combination of prime exponents bounded by those in the original magnitude. The combinations are generated completely and sorted from smallest to largest. The divisor count and sum accompany the list. MathWorld’s Divisor Function reference distinguishes the number of positive divisors from their sum.
Keep the original signs and omit thousands separators, decimal points and scientific notation.
Review the named primary result and open calculation details to check the arithmetic.
Factors of 36: worked example
The square preset enters 36. Its positive divisors are 1, 2, 3, 4, 6, 9, 12, 18, 36. The count is 9 and their sum is 91.
1 × 36 = 36
2 × 18 = 36
3 × 12 = 36
4 × 9 = 36
6 × 6 = 36
The prime-power decomposition is 2^2 × 3^2. Prime factors are the prime building blocks; the full list also contains one and composite divisors.
Why a square has one middle factor pair
Every positive factor pair has a smaller member no greater than the square root of the magnitude. Generate the pair from that smaller member and avoid repeating the reverse order. When the magnitude is a square, the equal middle pair appears once. The worked example’s list and pair checks come from the same complete prime decomposition.
The unit preset enters 1 and produces its only positive divisor, 1. Its classification is neither prime nor composite. Prime inputs have only one and themselves as positive divisors. These special cases belong in the same integer tool rather than separate fixed-number pages.
To list common divisors of several integers, first calculate their greatest common factor, then enter that nonzero result here. Every common positive divisor divides the GCF, and every positive divisor of the GCF divides the original integers. An all-zero list has infinitely many positive common divisors and cannot produce a finite complete list.
Negative factors, zero and complete-list limits
For a negative input, the main positive list describes the absolute value. Calculation details also show every signed integer divisor and the distinct signed pairs whose product equals the original negative input. The negative preset gives these pairs:
1 × -36 = -36
-1 × 36 = -36
2 × -18 = -36
-2 × 18 = -36
3 × -12 = -36
-3 × 12 = -36
4 × -9 = -36
-4 × 9 = -36
6 × -6 = -36
Zero is rejected: every nonzero integer divides zero, so a finite complete list cannot be returned. Symbolic expressions are a different factoring job and are not accepted by this tool.
Enter a nonzero integer with at most 14 digits, without decimal points, scientific notation or thousands separators. The complete list capacity is 20000 positive divisors. If that capacity were exceeded, the tool would report an error rather than display a partial list as complete. These are local computation limits; all returned factors, sums and products are exact.
Prime Factorization Calculator decomposes one integer into prime powers and repeated factors; Greatest Common Factor Calculator finds the greatest divisor shared by an integer list; Least Common Multiple Calculator finds the smallest positive shared multiple of nonzero integers, with an explicit zero convention.
When you need coefficients that reproduce the GCD, the Extended Euclidean Algorithm Calculator checks the original-input identity and its inverse residue.