Factors & Multiples

With building blocks you ask two questions: which small pieces make up this block, and which bigger shapes can it build? Whole numbers are the same: every number has a crew of factors and lives in an endless family of multiples.

Factors: the partners that divide exactly

If $a \times b = 12$ , then both $a$ and $b$ are factors of 12. List every multiplication that makes 12:

$12 = 1 \times 12 = 2 \times 6 = 3 \times 4$

So the factors of 12 are $1, 2, 3, 4, 6, 12$ — six of them. Factors come in pairs: finding 2 hands you 6 for free, and finding 3 hands you 4. To list all the factors of a number, test-divide from 1 upward and record both members of each pair.

Multiples: the times-table numbers

The multiples of 12 are its times table: $12, 24, 36, 48, \ldots$ — the list never ends.

Mind the direction

Factors search downward, multiples march upward: the largest factor of 12 is 12 itself, and there are finitely many; the smallest multiple of 12 is 12 itself, and they run on forever. Every number is both a factor and a multiple of itself.

Multiplication builds multiples; division tests factors — this pair of ideas is one inverse relationship seen from both sides.

Shared factors, shared multiples

Two numbers' factor sets and multiple lists overlap, and the overlap is where the useful players live:

Factors of 12: 1, 2, 3, 4, 6, 12; factors of 18: 1, 2, 3, 6, 9, 18. Shared: 1, 2, 3, 6 — the biggest, 6, is the greatest common factor. To cut two classes of 12 and 18 students into equal-sized teams, the team size must be a common factor of 12 and 18;
Multiples of 4: 4, 8, 12, 16, …; multiples of 6: 6, 12, 18, …. The first to appear together is 12, the least common multiple. If one event repeats every 4 days and another every 6, they next coincide 12 days later.

Long names?

Greatest common factor is often shortened to GCF (or GCD), least common multiple to LCM. Same ideas, shorter tags.

Divisibility shortcuts

No dividing needed — the digits give it away:

2: the last digit is 0, 2, 4, 6 or 8;
5: the last digit is 0 or 5;
10: the last digit is 0;
3: the digit sum is a multiple of 3. For 234: $2+3+4=9$ , so 234 is divisible by 3;
9: the digit sum is a multiple of 9. For 738: $7+3+8=18$ , and since 18 is a multiple of 9, so is 738.

A number with exactly two factors, 1 and itself, is prime — the indivisible piece in the factor world. Point the checker below at any number you like:

InteractiveFactor Checker

❌ 60 is not prime

Prime factorization：60 = 2² × 3 × 5

Primes up to 60

235711131719232931374143475359…

Check yourself

Quick quiz

1. Which list is exactly the full set of factors of 12?
2. The greatest common factor of 12 and 18 is…
3. The least common multiple of 4 and 6 is…