Divisibility, Factors, and Multiples

When does a number divide another?

Number theory is the study of the integers — the whole numbers …, -2, -1, 0, 1, 2, … — using nothing but the four basic operations. The first idea is [[divisibility|divisibility]]. We say that an integer a divides an integer b when b can be split into equal groups of size a with nothing left over. In symbols, a divides b (written a | b) means there is an integer k with b = a·k.

So 3 | 12 because 12 = 3·4, but 3 does not divide 13, because no whole number times 3 lands exactly on 13. The vertical bar is a true-or-false claim about two numbers; do not confuse it with the fraction 3/12.

Factors and multiples: two views of the same fact

If a | b, we call a a factor (or divisor) of b, and we call b a [[multiple|multiple]] of a. They describe the same relationship from opposite ends: 4 is a factor of 12, and 12 is a multiple of 4. Listing all the factors of a number is just a hunt for every a that divides it.

All factors of 24:
  1 × 24 = 24
  2 × 12 = 24
  3 ×  8 = 24
  4 ×  6 = 24
So the factors are: 1, 2, 3, 4, 6, 8, 12, 24.

Notice factors come in PAIRS that multiply to 24.
You only test up to sqrt(24) ≈ 4.9, because
after that the pairs just repeat in reverse.

GCF and LCM in plain terms

Two numbers usually share several factors. The largest one they share is the [[greatest-common-factor|greatest common factor]], and the smallest positive number they both divide into is the [[least-common-multiple|least common multiple]]. These two ideas run through everything that follows — from reducing fractions to combining cycles.

1. List the factors of each number. For 12: {1,2,3,4,6,12}. For 18: {1,2,3,6,9,18}.
2. The common factors are {1,2,3,6}; the largest is 6, so GCF(12,18) = 6.
3. List multiples: 12,24,36,… and 18,36,…; the first shared one is 36, so LCM(12,18) = 36.
4. Sanity check: GCF × LCM = 6 × 36 = 216 = 12 × 18. This product rule always holds.