Factors & multiples

1Summary 重點摘要

Factors of 6: 1,2,3,6; multiples of 3: 3,6,9...

2Vocabulary cards Vocabulary

3Step by step Steps

1. Part 1: What a factor is6 = 2 × 3, so both 2 and 3 are factors of 6; this is the definition: any two numbers whose product gives a number are its factors. Finding factors asks: which numbers divide it exactly? For example 6 ÷ 2 = 3 with no remainder, so 2 is a factor of 6; but 6 ÷ 4 has a remainder, so 4 is not. Take care that 1 is a factor of every number, since any number times 1 is itself; when listing factors, never leave out the pair 1 and the number itself.
2. Part 2: Pair up to find them allFind the factors of 12 in pairs: 1 × 12, 2 × 6 and 3 × 4 all make 12 exactly, so the factors are 1, 2, 3, 4, 6, 12. Why work in pairs? Each factor found reveals its partner at the same time, doubling the efficiency. At 5, 12 ÷ 5 has a remainder, and after that the pairs start repeating, so you can stop. Take care that repetition is the signal that the list is complete; trying 6 after finishing 4 only meets 6 again, so once repeats appear, stop instead of pressing on.
3. Part 3: What a multiple isMultiples of 3 come from multiplying 3 by 1, 2, 3 in turn: 3, 6, 9, 12, 15. It is like skip counting, adding 3 each time with no end, so every number has endless multiples. Neighbouring multiples are always the same distance apart, exactly 3, which is the rule of their order. Take care that a multiple is never smaller than the number itself; the smallest multiple of 3 is 3, because 3 × 1 = 3, so begin the list with the number itself and continue one after another in order.
4. Part 4: Factors finite, multiples infinite12 has only six factors, 1, 2, 3, 4, 6, 12, and you can count them all; but the multiples of 12, such as 12, 24, 36, never stop, and there is always a next one. Remember this point: factors are finite, multiples are infinite. Why are factors finite? No factor exceeds the number itself, so the search has a boundary and can be finished. Take care that tests often ask how many factors 12 has; the answer must list all six, and missing one counts as wrong, while questions on multiples only ever ask for the first few, so the two kinds of question make different demands.
5. Part 5: How factors and multiples connect12 ÷ 3 = 4 with no remainder says two things at once: 3 and 4 are factors of 12, and 12 is a multiple of both 3 and 4. Factors and multiples are two sides of one fact, seen from two angles in a single sentence. Why can one sentence be read two ways? Exact division is itself a two-way description: the divisor and the quotient together rebuild the dividend. Take care to read which is asked; asked for factors, answer 3 and 4; asked for multiples, answer 12, and the two lists must never be written for each other.
6. Part 6: Quick divisibility checksTo test whether 7 divides 56, think 7 × 8 = 56, so it divides exactly; reversing through multiplication is a quick route. To test 5, look at the last digit: only 0 or 5 works, like 60 yes and 62 no. To test 2, the last digit must be an even number. The reason behind these tricks is that both 5 and 2 decide divisibility by the ones digit alone. Take care that 3 is different: add all the digits and check the sum, as in 42, where 4 + 2 = 6 divides by 3, so 42 divides by 3 as well.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Factors divide exactly; multiples are the number times something
• Pair up to list factors: 12 = 1×12 = 2×6 = 3×4
• Factors are finite; multiples are infinite
• No remainder in 12 ÷ 3 → 3 is a factor, 12 a multiple
• Last digit 05 divides by 5; even last digit divides by 2