What Are Factors?

A factor of a number is any whole number that divides into it exactly — with no remainder. Factors always come in pairs, and every number has at least two factors: 1 and itself.

The word itself comes from the Latin factor, meaning “doer” or “maker” — fitting, since two factors multiplied together make the original number. Long before that word existed, Euclid described exactly this relationship in his Elements (c. 300 BCE), calling a number that divides another exactly a “measure” of it. The idea shows up constantly outside the classroom too: a warehouse manager arranging 60 boxes into equal rows, a designer choosing tile sizes that fit a wall exactly, or a programmer picking an array size that splits evenly across worker threads are all, whether they realise it or not, working with factors.

The Key Test

A is a factor of B if B ÷ A leaves remainder 0.

Examples

Factors of 6

6 ÷ 1 = 6 ✓ 6 ÷ 2 = 3 ✓ 6 ÷ 3 = 2 ✓ 6 ÷ 4 = 1.5 ✗ 6 ÷ 5 = 1.2 ✗ 6 ÷ 6 = 1 ✓

Factors of 6: 1, 2, 3, 6

Factors of 20

Test 1 to 20: 20 ÷ 1 ✓ ÷2 ✓ ÷3 ✗ ÷4 ✓ ÷5 ✓ ÷6 ✗ ÷7 ✗ ÷8 ✗ ÷9 ✗ ÷10 ✓ ÷11–19 ✗ ÷20 ✓

Factors of 20: 1, 2, 4, 5, 10, 20

Important Properties

Property	Example
1 is a factor of every number	1 ÷ 1 = 1, 7 ÷ 1 = 7
Every number is a factor of itself	15 ÷ 15 = 1
Factors are always less than or equal to the number	Factors of 12 are all ≤ 12
Factors come in pairs	1×12, 2×6, 3×4

Factors vs Multiples

Students often confuse these two ideas. A simple way to remember: factors are smaller (they go into the number); multiples are larger (the number goes into them).

Idea	Factors of 8	Multiples of 8
Direction	Divide INTO 8	8 multiplied by something
List	1, 2, 4, 8	8, 16, 24, 32, 40...
Count	Finite (always ends)	Infinite (never ends)

Key Takeaways

A factor divides a number with remainder zero.
Every number has 1 and itself as factors.
Factors are finite; multiples are infinite.
Knowing factors is essential for simplifying fractions and finding GCF.

Practice: Factors

Factor Question