Factors, Multiples & Primes
What Are Multiples?
A multiple of a number is what you get when you multiply that number by any whole number (1, 2, 3, 4, ...). Multiples are like the number's times table — they go on forever.
Multiplication tables — which are really just organised lists of multiples — are some of the oldest surviving mathematical documents in existence. Clay tablets from the Old Babylonian period, dated to around 1800 BCE, record tables of multiples for the base-60 number system the Babylonians used, the very same system that still gives us 60 minutes in an hour and 360 degrees in a circle today. The habit of listing out a number’s multiples to spot patterns is, in other words, nearly 4,000 years old.
Formal Definition
M is a multiple of N if M = N × k for some positive whole number k.
Examples
| Number | First 8 Multiples |
|---|---|
| 3 | 3, 6, 9, 12, 15, 18, 21, 24 |
| 7 | 7, 14, 21, 28, 35, 42, 49, 56 |
| 12 | 12, 24, 36, 48, 60, 72, 84, 96 |
| 0.5 | 0.5, 1, 1.5, 2, 2.5, 3, 3.5, 4 |
Key Properties of Multiples
- The first multiple of any number is the number itself (n × 1 = n).
- Multiples are always equal to or greater than the number.
- There are infinitely many multiples of any number.
- Every number is a multiple of 1.
- Zero is technically a multiple of every number (n × 0 = 0) — but is usually excluded from lists.
Multiples vs Factors – Side by Side
| Property | Factors | Multiples |
|---|---|---|
| Relationship | Divide INTO the number | Number divides INTO them |
| Size | ≤ the number | ≥ the number |
| Count | Finite | Infinite |
| Example (6) | 1, 2, 3, 6 | 6, 12, 18, 24... |
Multiples in Real Life
- Clocks: Minutes are multiples of 1; quarter hours are multiples of 15.
- Music: Beats repeat in multiples of the time signature.
- Packing: Boxes of 12 — stock levels are multiples of 12.
Key Takeaways
- Multiples = the number × 1, 2, 3, 4... (times table extended infinitely).
- A number is a multiple of its own factors.
- Multiples are always ≥ the original number (excluding 0).
- Finding common multiples leads to the LCM.