Factor Calculator
Enter a positive integer to find all of its factors, prime factorization, and key properties such as whether it is prime, perfect, or abundant.
Prime Factorization Calculator
Break down any positive integer into its prime factors using a step-by-step factor tree method with exponential notation.
Common Factors of Two Numbers
Enter two positive integers to find all factors they share in common, including the Greatest Common Factor.
Factor Pairs Calculator
Find all factor pairs of a number, where each pair of integers multiplies together to give the original number.
What is a Factor?
A factor of a number is any integer that divides that number exactly, leaving no remainder. Every positive integer has at least two factors: 1 and itself. Numbers with exactly two factors are called prime numbers.
12 / 1 = 12 (factor)
12 / 2 = 6 (factor)
12 / 3 = 4 (factor)
12 / 4 = 3 (factor)
12 / 6 = 2 (factor)
12 / 12 = 1 (factor)
Factors of 12: 1, 2, 3, 4, 6, 12
How to Find All Factors of a Number
To find all factors, divide the number by each integer from 1 up to its square root. When a division is exact, both the divisor and the quotient are factors. This method ensures no factors are missed while checking only the necessary range.
Square root of 36 = 6, so check divisors 1 through 6.
36 / 1 = 36 — pair: (1, 36)
36 / 2 = 18 — pair: (2, 18)
36 / 3 = 12 — pair: (3, 12)
36 / 4 = 9 — pair: (4, 9)
36 / 5 = 7.2 — not exact, skip
36 / 6 = 6 — pair: (6, 6) — same number, count once
All factors: 1, 2, 3, 4, 6, 9, 12, 18, 36
Prime Factorization
Prime factorization expresses a number as a product of prime numbers only. Every integer greater than 1 has a unique prime factorization. The process involves dividing by the smallest prime repeatedly until the quotient is 1.
72 / 2 = 36
36 / 2 = 18
18 / 2 = 9
9 / 3 = 3
3 / 3 = 1
72 = 2^3 x 3^2
Perfect, Abundant, and Deficient Numbers
Numbers can be classified based on the sum of their proper factors (all factors except the number itself). A perfect number has proper factors that sum exactly to itself. An abundant number has proper factors that sum to more than itself. A deficient number has proper factors that sum to less than itself.
Abundant: 12 — proper factors: 1+2+3+4+6 = 16 (greater than 12)
Deficient: 9 — proper factors: 1+3 = 4 (less than 9)
Factor Pairs
A factor pair is a set of two positive integers that multiply together to give a specific number. Every factor has exactly one corresponding partner. Factor pairs are useful in area problems, rectangle dimensions, and algebraic factoring.
1 x 24 = 24
2 x 12 = 24
3 x 8 = 24
4 x 6 = 24