HCF & LCM

1Summary 重點摘要

Factors of 6&8: 1,2; LCM: 24.

2Vocabulary cards Vocabulary

3Step by step Steps

1. Part 1: Finding common factorsList the factors of 6 and 8 separately: factors of 6 are 1, 2, 3, 6; factors of 8 are 1, 2, 4, 8. Only 1 and 2 appear on both lists — the common factors — and the largest of them, 2, is the highest common factor, or HCF. List factors in pairs starting from 1 — for example 6 = 1 × 6 = 2 × 3 — and nothing will be missed.
2. Part 2: Finding common multiplesWrite out the multiples of 4 and of 6: 4, 8, 12, 16, 20, 24 and 6, 12, 18, 24. The numbers appearing on both lists, 12 and 24, are common multiples, and the smallest of them, 12, is the lowest common multiple, or LCM. List multiples by multiplying 1, 2, 3 in turn, and stop as soon as the same number shows on both lists.
3. Part 3: HCF versus LCMThe HCF is the largest of the common factors, never bigger than the smaller of the two numbers. The LCM is the smallest of the common multiples, never smaller than the larger number. Take 4 and 6: HCF = 2 and LCM = 12 — one very small, one quite large. Remember these two range rules and check every answer against them; anything outside the range betrays a mistake at once.
4. Part 4: The HCF simplifies fractionsTo bring 812 to simplest form: the common factors of 8 and 12 are 1, 2 and 4, so the HCF is 4. Divide top and bottom by 4: 8 ÷ 4 = 2 and 12 ÷ 4 = 3, giving 23 directly. Using the HCF reaches simplest form in one step, not several. After simplifying, check whether top and bottom still share a factor; none left means simplest form.
5. Part 5: The LCM converts fractionsFor 14 + 16, find the lowest common multiple of 4 and 6 to use as the common denominator: LCM = 12. Then 14 = 312 and 16 = 212, and adding gives 512. Converting with the LCM keeps every number as small as possible. Use 24 instead and 624 + 424 = 1024 still needs reducing — proof of how well the LCM serves.
6. Part 6: Telling the contexts apartPack 18 sweets and 24 biscuits into gift bags with exactly the same contents; how many bags at most? That asks for HCF(18, 24) = 6. Two buses leave together; after how many minutes do they leave together again? That asks for the LCM. Equal sharing uses HCF; meeting again uses LCM. The key question to ask first: must the items be used up, or must an event recur together?

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Common factors are shared; the biggest is the HCF
• Common multiples are shared; the smallest is the LCM
• HCF simplifies fractions; LCM converts them
• List, circle the shared ones, pick biggest/smallest
• Most equal groups → HCF; next coincidence → LCM