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Chapter 3 · Finding Common Ground

What HCF and LCM do for consecutive, even, and co-prime numbers

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What to assume they know

What they should be able to do

  • Describe, without listing, which pairs have an HCF equal to one of the pair itself, and do the same for the LCM
  • Write such a pair using a letter
  • State and defend a general claim about the HCF of two consecutive numbers, two consecutive even numbers, two consecutive odd numbers, two even numbers, and two co-prime numbers
  • State and defend a general claim about the LCM of two multiples of 3, two consecutive even numbers, two consecutive numbers, and two co-prime numbers
  • Predict the effect on the HCF of doubling both numbers, and explain it from the factorisations
  • Given two numbers written as a shared multiplier times something, decide whether that multiplier is the HCF
  • Say what has to be true of the two leftover multipliers for the shared multiplier to be the whole answer

Where it usually goes wrong

  • "If both numbers are multiples of 14 then 14 is their HCF." The printed counterexample is 14 × 6 and 14 × 9. The shared multiplier is a common factor always; it is the highest one only when what is left over shares nothing. This is the single most important correction in the topic, and the chapter builds the whole passage around it.
  • "Two even numbers have HCF 2." They have an even HCF. 12 and 18 give 6; 20 and 30 give 10. The precise version — exactly 2 — belongs to two consecutive even numbers, and the difference between those two statements is what the exercise is testing.
  • "Consecutive numbers might share a factor if they are big enough." Size is irrelevant. Any common factor would have to fit into the gap between them, and the gap is 1. State it that way — but note the chapter asks the reader to find the reason rather than supplying it, so present the argument as reasoning being done, not as a quotation.
  • "Doubling both numbers doubles the LCM too." It does, but do not smuggle it in — the book asks only about the HCF here. If the explanation raises it, it must raise it as its own extension and check it on the printed pair: 270 and 50 against 540 and 100.
  • "A pattern that works on my three examples is a fact." This section is training the opposite reflex. The instruction attached to every one of its cases is to look for a reason, not merely for more agreeing examples.
  • "The letter n makes it harder." It is what turns "I checked six pairs" into "this holds for all pairs". The chapter introduces algebra here for exactly that reason and says so.

Questions to check understanding

  • State the HCF or the LCM of a described kind of pair rather than of two given numbers
  • Describe, using a letter, every pair whose HCF is one of the pair itself
  • Judge a stated general claim about HCF or LCM and, if it fails, produce a counterexample
  • Given two numbers as products sharing a visible factor, find their HCF
  • Predict the effect on the HCF or the LCM of scaling both numbers
  • Find two numbers meeting stated HCF and LCM conditions — the closing exercise asks for a pair with HCF 1 and LCM 66 (Part II, p.63, question 4)
  • Decide where the LCM of two different primes must sit relative to the numbers and their product (Part II, p.64, question 10)

Examples worth working on the board

  • The opening pair (Part II, §3.3, p.58). 6 and 18, whose HCF is 6. The reader is asked to find more pairs like it and then to say what such pairs have in common. The book's own answer to the second question: whenever either number divides the other, so that one is a multiple of the other.
  • The cartoon on p.58 (Part II, §3.3). Checked against the printed page. A large orange 18 and a large yellow 6, drawn with arms and legs, playing football on a green field, with a speech bubble offering a deal about being each other's factor and multiple. It is the section's motto and worth one beat; redraw it.
  • The algebra (Part II, §3.3, pp.58–59). Inputs exactly as printed: take n and 5n; their HCF is then n itself. Two follow-up prompts: given m, what could the partner be; given 7k, what could the partner be. Neither is answered in the book.
  • The HCF cases to generalise (Part II, §3.3, "Figure it Out", p.59). Five kinds of pair: two consecutive even numbers; two consecutive odd numbers; two even numbers; two consecutive numbers; two co-prime numbers. The instruction is to try examples first, then state something general, then look for a reason. Two cautions on the wording of the general statements. For two even numbers the honest claim is that the HCF is even — not that it is 2; 12 and 18 have HCF 6. For two consecutive even numbers the HCF is exactly 2, and that is a genuinely stronger claim than the previous one.
  • The LCM cases to generalise (Part II, §3.3, "Figure it Out", p.59). First, 3 and 24, whose LCM is 24, with the same two follow-ups: find more such pairs, then describe them with a letter. Then four kinds of pair: two multiples of 3; two consecutive even numbers; two consecutive numbers; two co-prime numbers. Caution: for two multiples of 3 the safe claim is that the LCM is again a multiple of 3. It is not in general 3 times anything predictable — 6 and 9 have LCM 18, while 6 and 12 have LCM 12.
  • Doubling (Part II, §3.3, pp.59–60). Checked against the printed page. Inputs as the book prints them: 270 = 2 × 3 × 3 × 3 × 5 and 50 = 2 × 5 × 5, with the shared 2 and 5 boxed in colour and the HCF reported as 10; then the doubled pair, 540 = 2 × 2 × 3 × 3 × 3 × 5 and 100 = 2 × 2 × 5 × 5, with the extra 2 boxed in as well and the HCF reported as 20.
  • The trap (Part II, §3.3, p.60). Checked against the printed page. The two numbers are printed as products, not evaluated: 14 × 6 and 14 × 9. Beneath each the book writes the primes — 2 × 7 × 2 × 3 and 2 × 7 × 3 × 3 — with the 2 × 7 in each underbraced in red and tagged 14, and the surviving shared 3 ringed in green. Its reported HCF: 14 × 3 = 42. This picture is the heart of the topic; give it time and let the second shared 3 arrive as a surprise.
  • The follow-up pairs (Part II, §3.3, p.60). Four pairs, all written as products so the shared multiplier is visible: 18 × 10 and 18 × 15; 10 × 38 and 10 × 21; 5 × 13 and 5 × 20; 12 × 16 and 12 × 20. The book states that in the second of these the HCF is 10, and asks in which cases the HCF equals the shared multiplier and when that happens. It does not answer. What: it happens exactly when the two leftover multipliers have nothing in common.

Figures to have open

  • Two factorisation rows with independently boxable primes, reused from HCF: take the fewest occurrences of each prime, and able to show a product form (14 × 6) collapsing into its primes.
  • A number strip on which two adjacent positions can be marked and the gap between them measured, for section 4.
  • Two dials setting the multipliers either side of a fixed shared factor, with a live HCF readout, for section 11. This is the only genuinely interactive figure the topic needs and it carries the payoff.
  • The football cartoon of 18 and 6 is the book's own art; redraw it or drop it, but do not reproduce it.
  • No numbered figure exists to cite: this chapter prints no "Fig. 3.x" label anywhere on pp.47–66 (all twenty pages checked).

Where this sits in the book

The book

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