Finding Common Ground · Lesson 5 of 7
Patterns, General Statements, and Counterexamples
“Explore what HCF and LCM reveal about special pairs of numbers.”
• Recognise when one number is the HCF or LCM of a pair. • Make justified statements about consecutive and coprime numbers. • Explain the effect of doubling both numbers on HCF. • Test a proposed generalisation with examples and counterexamples.
When the answer is one of the numbers
The HCF of 6 and 18 is 6, while the LCM of 3 and 24 is 24. In each pair, one number divides the other. If a positive number n divides kn, where k is a positive whole number, n is already the largest possible common factor and kn is already the smallest possible common multiple. This turns observed examples into a general statement.
Problem
Find the HCF and LCM of 8 and 40 without a factorisation table.
- 1.40 = 5 × 8, so 8 divides both and is the greatest factor of 8.
- 2.40 is already a multiple of 8 and 40.
- 3.HCF = 8 and LCM = 40.
Look carefully at kinds of pairs
Two consecutive positive whole numbers have no common factor greater than 1: any common factor would divide their difference, which is 1. Thus their HCF is 1 and their LCM is their product. Consecutive odd numbers differ by 2, but an odd common factor cannot be 2, so they also have HCF 1. Consecutive even numbers can be written 2n and 2(n + 1). After their shared factor 2 is removed, n and n + 1 have HCF 1, so the original HCF is exactly 2. Merely saying “two even numbers” guarantees at least 2, not an exact HCF.
A statement about a pattern that is intended to hold for every case in a specified group, supported by an explanation rather than only a few examples.
| Pair type | What is certain? | What is not fixed? |
|---|---|---|
| Consecutive numbers | HCF = 1; LCM = product | The numerical product varies |
| Consecutive even numbers | HCF = 2; LCM = product ÷ 2 | Their LCM depends on the pair |
| Any two even numbers | HCF is at least 2 | HCF can be greater than 2 |
| Coprime numbers | HCF = 1; LCM = product | Neither number has to be prime |
| Two multiples of 3 | HCF and LCM are multiples of 3 | Neither has a single fixed value |
Problem
Compare the HCFs of 6 and 8, and of 8 and 12.
- 1.6 and 8 share factors 1 and 2, so HCF = 2.
- 2.8 and 12 share 1, 2 and 4, so HCF = 4.
- 3.Both pairs are even, but “HCF = 2 for every even pair” is false.
Scaling and shared pieces
Doubling both numbers adds one factor of 2 to both prime factorisations, so it doubles their HCF. For 270 and 50, HCF = 10; for 540 and 100, HCF = 20. More generally, multiplying both positive numbers by the same positive whole number multiplies their HCF by that number. But a visible common multiplier need not be the complete HCF: 14 × 6 and 14 × 9 also share a factor 3 from 6 and 9.
Problem
Find the HCF of 14 × 6 and 14 × 9.
- 1.The numbers are 84 and 126, so both have the obvious factor 14.
- 2.The remaining factors 6 and 9 share 3.
- 3.HCF = 14 × 3 = 42, rather than 14.
Several agreeing examples suggest a conjecture. Explain why it holds for every pair in the stated class, or find one counterexample that shows the proposed statement is too broad.
Quiz
The HCF of 15 and 16 is
For positive n, what is the LCM of n and 5n?
Which statement about any two even numbers is always true?
If both numbers are doubled, their HCF
What is the HCF of 14 × 6 and 14 × 9?
Practice Problems
- Describe all positive pairs for which one number is both a factor of the other and their HCF.
- Explain why consecutive numbers are coprime, then find the HCF and LCM of 34 and 35.
- Investigate two consecutive odd numbers and justify their HCF.
- Find two pairs of even numbers with different HCFs to disprove “every even pair has HCF 2.”
- Find the HCF of 18 × 10 and 18 × 15. Is 18 the complete HCF?
- Double 84 and 126; predict and then verify the new HCF.
Key Takeaways
• If one positive number divides the other, the smaller is HCF and the larger is LCM. • Consecutive numbers have HCF 1 and LCM equal to their product. • Any two even numbers have an even HCF, but its exact value varies. • Doubling both numbers doubles their HCF. • A displayed common multiplier may not capture another factor shared by the remaining parts.