The short answer
A factor is a whole number that divides another number with no remainder. 24 has eight factors: 1, 2, 3, 4, 6, 8, 12, and 24. Every composite number breaks down into a unique set of prime factors — its prime factorization — while a number with exactly two factors (1 and itself) is prime.
Key takeaways
- Factors always come in pairs that multiply to the original number (1×24, 2×12, 3×8, 4×6), except for perfect squares, which have one unpaired factor.
- You only need to test divisors up to the square root of a number to find every factor — anything beyond that has already shown up as a pairing partner.
- Every composite number has exactly one prime factorization (the Fundamental Theorem of Arithmetic) — the same set of primes no matter how you break it down.
- A number is prime only if it has exactly two factors, 1 and itself. The number 1 itself is neither prime nor composite.
Finding factors efficiently: the square root shortcut
Checking every number from 1 to n to find factors works, but it's slow for large numbers. The faster method only tests divisors up to √n: whenever a divisor d divides n evenly, its pair n ÷ d is automatically a factor too. For 36 (√36 = 6), testing just 1 through 6 finds every factor: 1↔36, 2↔18, 3↔12, 4↔9, 6↔6 — the last pair collapses into a single factor since 6 × 6 = 36, which is why perfect squares always have an odd factor count.
Full factor list vs. prime factorization
| Concept | What it is | Example (for 24) |
|---|---|---|
| Full factor list | Every divisor of the number | 1, 2, 3, 4, 6, 8, 12, 24 |
| Prime factorization | The number as a product of primes | 2³ × 3 |
| Factor pairs | Two factors that multiply to the number | (1,24) (2,12) (3,8) (4,6) |
There's a direct link between the two: if you write the prime factorization as p₁^a × p₂^b × ..., the total factor count is (a+1)(b+1).... For 24 = 2³ × 3¹, that's (3+1)(1+1) = 8 — exactly matching the 8 factors listed above.
Worked example: factoring 24
24 ÷ 2 = 12
12 ÷ 2 = 6
6 ÷ 2 = 3
3 ÷ 3 = 1 (stop — 3 is prime)
24 = 2 × 2 × 2 × 3 = 2³ × 3
Dividing repeatedly by the smallest possible prime at each step guarantees you land on the correct prime factorization, and from there the (a+1)(b+1) rule confirms 24 has exactly 8 factors without listing them one by one.
Common mistakes to avoid
- Forgetting that 1 and the number itself always count as factors, even though they're easy to overlook when listing the "interesting" ones.
- Testing every number up to n instead of up to √n — for large numbers this is dramatically slower than necessary.
- Confusing "factor" with "multiple" — a factor divides into the number, while a multiple is what you get by multiplying the number by an integer.
- Assuming every number has an even number of factors — perfect squares always have an odd count, since one factor pairs with itself.
Related calculators
- Prime Factorization Calculator — focus purely on breaking a number down into its prime building blocks.
- Common Factor Calculator — compare factor lists across two or more numbers at once.
- Greatest Common Factor Calculator — jump straight to the largest shared factor between numbers.
- Least Common Multiple Calculator — go the other direction and find shared multiples instead of factors.