Factors vs Multiples: The Difference Using 12 and 36

Factors divide; multiples extend

A positive factor of 12 divides 12 exactly. Its complete list is 1, 2, 3, 4, 6 and 12. A positive multiple of 12 is 12 times a positive whole number: 12, 24, 36, 48 and so on. There are finitely many positive factors of 12 but infinitely many positive multiples.

Twelve itself belongs to both lists. It divides itself, and it is 12 × 1. That overlap does not make the words interchangeable.

Read one equation in two directions

The equation 12 × 3 = 36 says that 12 and 3 are factors of 36. It also says that 36 is a multiple of both 12 and 3. Notice which number the question starts with before choosing a list.

To arrange 36 tiles in equal rows, use its factor pairs: 1 × 36, 2 × 18, 3 × 12, 4 × 9 and 6 × 6. To buy packets containing 12 tiles each, use multiples of 12 to find possible totals. The context tells you which relationship is useful.

Check an answer with a remainder

Thirty is not a multiple of 12 because 30 ÷ 12 is 2 with remainder 6. Equivalently, 12 is not a factor of 30. By contrast, 36 ÷ 12 has remainder zero.

Number JD lists positive factors and the first twelve positive multiples on each number page. Zero and negative multiples exist mathematically, but they are outside those explicitly labeled lists.

Check the mathematical definitions

For fractions, percentages and worked arithmetic, see OpenStax Prealgebra. For number definitions, see OpenStax on prime and composite numbers and Wolfram MathWorld on digital roots. Worked examples on this page are calculated for this guide.