Factors & Multiples
Two words that get mixed up constantly, and the difference in one sentence.
The explanation
A factor of 12 is any whole number that divides into 12 with nothing left over: 1, 2, 3, 4, 6, 12. Factors are always less than or equal to the number.
A multiple of 12 is what you get by multiplying 12 by a whole number: 12, 24, 36, 48, and on forever. Multiples are always greater than or equal to the number.
An easy way to keep them straight: factors *fit inside*, multiples *march outward*. Every number has a limited list of factors and an unlimited list of multiples.
For integers a and b, we say a is a factor (or divisor) of b when there exists an integer k with b = a·k. The same equation read the other way says b is a multiple of a. Factor and multiple are two names for the two sides of one divisibility relation.
Finding all factors efficiently uses pairing: factors come in pairs whose product is n, so you only need to test divisors up to √n. For 36 you test 1, 2, 3, 4, 5, 6 and pick up 36, 18, 12, 9, and 6 as the partners. The pair collapses to a single value exactly when n is a perfect square, which is why perfect squares have an odd number of factors.
Worked example
List every factor of 36.
- Test divisors up to √36 = 6.
- 1×36, 2×18, 3×12, 4×9, 6×6.
- 6 pairs with itself, so it is listed once.
Answer: 1, 2, 3, 4, 6, 9, 12, 18, 36 — nine factors, odd because 36 is a perfect square.
Common mistakes
- Swapping the words: saying 24 is a factor of 12. It is the other way round.
- Forgetting that 1 and the number itself are always factors.