What is a factor?
A factor of a number is any whole number that divides it evenly, with nothing left over. The factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24 because each divides 24 evenly with no remainder. Every number is a factor of itself, and 1 is a factor of every number.
Factor pairs
A factor pair is a pair of factors whose product equals the original number. For 24: 1×24, 2×12, 3×8, and 4×6 all equal 24. Factor pairs are just the plain list of factors grouped two at a time, and they naturally stop once you reach the square root of the number — past that point you're just seeing the same pairs in reverse.
Prime factorization
A number's prime factorization expresses it as a product of prime numbers. 24 = 2 × 2 × 2 × 3, usually written more compactly as 2³ × 3. Every whole number greater than 1 has exactly one prime factorization, regardless of which order you find the prime factors in — a fact important enough to have its own name, the fundamental theorem of arithmetic.
Reading the factor tree
A factor tree shows the prime factorization visually: starting from the number itself, each composite number branches into two factors, and those branch further until every branch ends in a prime — shown circled in blue. Multiply all the circled primes together and you get back the original number, regardless of which factor pairs were chosen along the way.
Frequently asked questions
What's the difference between a factor and a prime factor?
A factor is any number that divides evenly into another — 24 has eight factors. Prime factorization is the process of breaking a number down into the prime numbers that multiply together to make it. For example, 24 = 2 × 2 × 2 × 3, or 2³ × 3.
Is 1 a prime number?
No. 1 is neither prime nor composite — it's its own special case. It has exactly one factor (itself) and no prime factorization at all, which is why the factor tree for 1 is just a single node with nothing to branch into.
Can a factor tree look different for the same number?
The shape can differ depending on which factor pair you start with, but the prime numbers at the tips of the branches are always the same, with the same multiplicities — that's the fundamental theorem of arithmetic in action.