Factors divide into a number; multiples are built from it. Two directions, endlessly confused with each other — this guide separates them for good.
Factors and multiples describe two opposite relationships between numbers. A factor divides evenly into a number. A multiple is what you get when you multiply a number up. This guide covers both in full — including how to classify factors as prime or composite, and how to find multiples numbers share in common.
A factor of a number is any number that divides into it exactly, with no remainder. If our Divisibility Rules article tells you whether one number divides another, factors are simply the complete list of every number that does.
4 is a factor of 12, because 12 ÷ 4 = 3 exactly. 5 is not a factor of 12, because 12 ÷ 5 leaves a remainder.
The most reliable method is the pair method: test each number starting from 1, and whenever it divides evenly, record both it and its matching pair.
1×24, 2×12, 3×8, 4×6 — after 4, the next number to test is 5, which doesn't divide evenly, and testing further would only repeat pairs already found in reverse. All factors of 24: 1, 2, 3, 4, 6, 8, 12, 24 (8 factors total).
You only need to test numbers up to the square root of the target number — once your test number passes the square root, every remaining factor has already been found as part of an earlier pair.
1 is a factor of every number, since every number divides evenly by 1. And every number is a factor of itself, since any number divides evenly into itself exactly once. These two facts are always true, no matter which number you're factoring.
A prime factor is a factor that is also a prime number — a factor with exactly two factors of its own (1 and itself). Among all the factors of a number, only some are prime.
The factors of 30 are 1, 2, 3, 5, 6, 10, 15, 30. Checking each one: 2, 3, and 5 are prime. So the prime factors of 30 are 2, 3, and 5.
A composite factor is a factor that is itself composite — meaning it has more than two factors of its own. Every factor of a number is either 1, prime, or composite; there's no fourth option.
The factors of 30 are 1, 2, 3, 5, 6, 10, 15, 30. Removing 1 and the primes (2, 3, 5) leaves the composite factors: 6, 10, 15, and 30.
This mixes up "prime factors" (a subset) with "all factors" (the complete list).
The fix: every factor list contains three kinds of members: the number 1 (neither prime nor composite), prime factors, and composite factors. Only a small handful of factors are ever prime — most factors of composite numbers are composite themselves.
Prime factorization means breaking a number down until only prime numbers remain, multiplied together. Every composite number has exactly one prime factorization — this is such a fundamental property of numbers that it has its own name: the Fundamental Theorem of Arithmetic.
36 → 4 × 9 → (2 × 2) × (3 × 3). Prime factorization of 36: 2 × 2 × 3 × 3 (only 2 and 3 appear, but 2 appears twice and 3 appears twice).
100 ÷ 2 = 50. 50 ÷ 2 = 25. 25 ÷ 5 = 5. 5 ÷ 5 = 1 — stop once you reach 1. Prime factorization of 100: 2 × 2 × 5 × 5.
Most numbers have an even number of factors, because factors pair up neatly (like 4 and 6 for 24). But perfect squares are the exception — they always have an odd number of factors, because one factor pairs with itself.
| Number | Factors | Count |
|---|---|---|
| 16 | 1, 2, 4, 8, 16 | 5 (odd — perfect square) |
| 25 | 1, 5, 25 | 3 (odd — perfect square) |
| 20 | 1, 2, 4, 5, 10, 20 | 6 (even — not a perfect square) |
In 16, the factor pair 4×4 only contributes a single 4 to the list (not two), which is exactly why the total count comes out odd.
Tap any number below to see its complete factor list, sorted into prime, composite, and neither.
Tap a number above to see its full factor breakdown.
A multiple of a number is what you get by multiplying it by a whole number. If factors divide into a number, multiples are built from a number by scaling it up.
12 is a multiple of 4, because 4 × 3 = 12. The first several multiples of 4 are: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40.
Multiples are found simply by skip-counting — the same repeated-addition idea covered in our Times Tables Trainer article.
| Number | First 10 Multiples |
|---|---|
| 4 | 4, 8, 12, 16, 20, 24, 28, 32, 36, 40 |
| 6 | 6, 12, 18, 24, 30, 36, 42, 48, 54, 60 |
| 7 | 7, 14, 21, 28, 35, 42, 49, 56, 63, 70 |
Unlike factors, which are always a finite, complete list, multiples never end — every number has infinitely many multiples, since you can always multiply by a bigger whole number.
A common multiple of two numbers is any value that appears in both of their multiple lists.
Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36...
Multiples of 6: 6, 12, 18, 24, 30, 36...
Common multiples of 4 and 6: 12, 24, 36, 48, 60... — this list is exactly as infinite as any other list of multiples.
| Factors | Multiples | |
|---|---|---|
| Direction | Divide into the number | Built by multiplying the number up |
| Quantity | Finite — a complete, countable list | Infinite — never ends |
| Size compared to the number | Always less than or equal to it | Always greater than or equal to it |
| Smallest value | 1 | The number itself |
This confuses a multiple (8 is built from 4) with a factor (which must divide into 4).
The fix: 8 is a multiple of 4 (4×2=8), not a factor of it. The factors of 4 are only 1, 2, and 4 — every one of them is smaller than or equal to 4, never larger.
Arranging 24 chairs into equal rows uses factors — only 1, 2, 3, 4, 6, 8, 12, or 24 rows will divide the chairs evenly with none left over. Two buses that leave a station every 4 minutes and every 6 minutes will next arrive together at a common multiple of 4 and 6 — 12 minutes later, then every 12 minutes after that.
The Fundamental Theorem of Arithmetic — the fact that every number has exactly one prime factorization — was proven by Euclid over 2,300 years ago and remains one of the cornerstones of all modern number theory.
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Factors and multiples are the direct foundation for the next two topics in this strand — HCF (highest common factor) and LCM (lowest common multiple) — both of which are essential for simplifying fractions, solving scheduling problems, and comparing quantities. Prime factorization specifically also underpins cryptography, the technology that keeps online banking and messaging secure.
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