Factors & multiples
The idea
Some numbers fit perfectly inside other numbers. A factor of 24 is any number that divides 24 with nothing left over — like 6, since 6 × 4 = 24. Multiples run in the other direction: the multiples of 6 are its skip-counting list, 6, 12, 18, 24, 30, and on forever. The two ideas are partners: saying 6 is a factor of 24 and saying 24 is a multiple of 6 describe the same neat fit.
Picture factors as ways to arrange objects in equal rows. You can lay out 24 juice boxes in 2 rows of 12 or 4 rows of 6, so 2, 12, 4, and 6 are all factors. Because rows and columns come together, factors arrive in pairs, and that makes hunting for them organized: test 1, 2, 3, and so on, writing down each partner. People often mix the two words up, so keep this anchor: a factor of a number is never bigger than the number itself, while its multiples are never smaller.
Worked example
You want to arrange 24 juice boxes into equal rows with none left over. Find every number of rows that works — in other words, find all the factors of 24.
- Test numbers in order and record each partner. One row of 24 works, so 1 and 24 form the first factor pair.
- Since 24 is even, 2 works: 2 × 12 = 24, giving the pair 2 and 12. Three rows work too, because 3 × 8 = 24.
- Keep going: 4 × 6 = 24, so 4 and 6 pair up. But 5 fails — 5 × 4 = 20 is too small and 5 × 5 = 25 is too big, so skip-counting by 5 never lands exactly on 24.
- The next number to try is 6, and 6 already appears in your list as the partner of 4. When the pairs start repeating, the hunt is over.
- List everything found, in order: 1, 2, 3, 4, 6, 8, 12, 24. That is every way to make equal rows.
Answer. The factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24 — eight possible row arrangements.
Check your understanding
- Why do factors almost always come in pairs, and which special numbers give a pair where both partners are the same?
- How do you know when you can safely stop testing numbers while hunting for factors?
- What is the difference between the factors of 12 and the multiples of 12, and which list goes on forever?
- How could skip-counting help you decide whether 7 is a factor of 42?