Finding Common Ground · Lesson 7 of 7
Chapter Summary and Practice
“Connect factors, multiples, prime pieces, HCF and LCM through mixed decisions and problems.”
• Choose HCF or LCM from the meaning of a situation. • Use prime factorisation and shared division with checks. • Explain patterns and the two-number product relationship. • Solve mixed applications and investigate a prime-factor colour puzzle.
What we have learned
A factor fits exactly inside a number; a multiple is obtained by multiplying it by a positive whole number. Prime factorisation exposes the pieces behind both relationships. HCF asks for the greatest piece shared by all the numbers. LCM asks for the smallest number that contains all their prime pieces. Decide which question the story asks before calculating.
| Idea | Meaning and method | Typical clue |
|---|---|---|
| Prime factorisation | Keep splitting or dividing until only primes remain. | Find all factors or compare prime pieces. |
| HCF | Use the fewest copies of each shared prime. | Largest identical tile, bag, cube or jump size. |
| LCM | Use the most copies of each prime found anywhere. | First common length or next time cycles meet. |
| Shared division | Left common-factor product gives HCF; include coprime remainders for LCM. | Find both efficiently. |
| Two-number identity | HCF × LCM = product of two positive numbers. | Check or recover one result from the other. |
The checks work in opposite directions: the HCF must divide each original number, while each original number must divide the LCM. For a factor–multiple pair, the smaller is HCF and the larger is LCM. For consecutive numbers, the HCF is 1. These observations can shorten a calculation, but they need a reason, not just a pattern noticed twice.
Problem
A 12 cm by 18 cm by 36 cm box must be filled with identical whole-centimetre cubes without gaps. What is the greatest cube edge?
- 1.The edge must divide all three dimensions, so this is an HCF problem.
- 2.12 = 2² × 3; 18 = 2 × 3²; 36 = 2² × 3². The least shared counts give 2 × 3 = 6.
- 3.A 6 cm cube fits: 2 × 3 × 6 = 36 such cubes fill the box.
Problem
A herd of fewer than 200 animals can be divided equally among 3 gates, then 5 gates, then 7 gates. What could its size be?
- 1.The size must be divisible by 3, 5 and 7, so it is a multiple of their LCM.
- 2.The primes are distinct, so LCM = 3 × 5 × 7 = 105.
- 3.The only positive multiple of 105 below 200 is 105.
Problem
Find an efficient common denominator for 1/15 and 1/20, then add.
- 1.15 = 3 × 5 and 20 = 2² × 5; their LCM is 2² × 3 × 5 = 60.
- 2.1/15 = 4/60 and 1/20 = 3/60.
- 3.Their sum is 7/60. The same idea scales to the five fractions in the chapter’s historical problem.
Think, check, and explore
The closing problems mix direct calculations with decisions: matching coloured-star cycles is LCM, the largest whole-number cube fitting a box is HCF, and a remainder condition adds a further check after finding a common multiple. The dog and rabbit gain a net 2 feet per paired leap, so a 150-foot head start takes 75 such leaps to erase. In the “Mystery Colours” panel, a prime receives one colour and a composite’s ring combines the colours of its prime factors, with repeated factors represented again. Test the rule on small numbers before extending the pattern to 101–110.
Do not choose HCF just because a question says “smallest number”: a smallest number divisible by several values calls for LCM. Do not reuse the two-number HCF × LCM identity with three numbers. A remainder condition must be checked after divisibility.
Quiz
The largest equal whole-number cube edge that fills a box signals
The first time two cycles coincide again signals
For 12 and 18, HCF × LCM equals
What is the HCF of two consecutive positive numbers?
Which check is appropriate for an LCM of a and b?
What is the smallest positive number divisible by 3, 5 and 7?
Practice Problems
- Two coloured star patterns repeat after 6 and 8 positions. At which next position do their starting colours coincide?
- Decide whether 5 × 7 × 11² is a factor of 5 × 7² × 11 × 2. Explain by comparing prime occurrences.
- Find HCF and LCM of 45 and 36 and verify their product relationship.
- Find two positive numbers with HCF 1 and LCM 66; explain your choice.
- Which proposed cube sides 9, 6, 4, 3 and 2 cm fit exactly into a 12 by 18 by 36 cm box?
- Find the smallest positive multiple of 3, 4, 5 and 7 that leaves remainder 10 on division by 11.
- How many paired leaps close a 150-foot gap if the pursuer gains 2 feet per paired leap?
- Find the smallest positive number divisible by 1, 2, 3, 4, 5, 6, 8, 9 and 10.
- Use the LCM of the denominators to add 8/15 + 1/20 + 7/36 + 11/63 + 1/21.
- Infer the colour pattern in the page-number rings from 1–100 and explain the ring for 102 or 105.
- Which of 36, 18, 3 and 2 is the largest number that divides both 306 and 36?
- A group divides equally into teams of 6 and into teams of 9, but not teams of 10. Which among 72, 90, 45, 3 and 36 could be its size?
- Explain the LCM of two distinct primes and whether it exceeds both primes.
Key Takeaways
• Prime factorisation supports both factor and multiple reasoning. • HCF takes the fewest shared prime pieces; LCM takes the most needed prime pieces. • Choose HCF for a greatest equal size and LCM for a first shared repeat or smallest common divisible number. • Verify HCF by dividing each original number by it; verify LCM by dividing it by each original number. • For two positive numbers, HCF × LCM = product; special-pair patterns can simplify calculations. • A conjecture needs a reason or a counterexample, and mixed problems may add conditions beyond HCF or LCM.
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Efficient Joint Methods and the Product Relationship
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