HCF: take the fewest occurrences of each prime

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What to assume they know

Breaking a number down to primes, and why the result is unique — producing a prime factorisation
Reading every factor of a number off its prime factorisation — a number's factors are exactly the subparts of its own factorisation, and nothing else
Class 6 Prime Time: common factors, and the Jump Jackpot game on a number line
Area of a rectangle as rows times columns, for the tiling problem

What they should be able to do

Recognise a "largest equal piece" problem and say why it is a common-factor problem
State what the highest common factor of two numbers is, and give the other name the chapter attaches to it
Explain why, in the tiling and packing problems, the largest common factor is the one wanted
List the common factors of two numbers by matching subparts of their two factorisations
Recognise the case where the only common factor is 1, and say what the factorisations look like when that happens
Compute the HCF directly by taking, for each shared prime, the smaller of its two counts
Extend the same procedure to three numbers
Explain why two composite splittings sharing nothing in common is no evidence at all about the HCF

Where it usually goes wrong

"If the two splittings have no piece in common, the numbers have no common factor." This is the exact error the p.54 question is built to expose. 6 × 12 and 8 × 18 share nothing on the surface, and 72 divides 144 exactly. Example 3 looks like the same argument and is valid only because both splittings there were into primes. Show the two side by side; the difference is the whole lesson.
"HCF means listing both sets of factors and picking the biggest match." That is how the chapter starts and precisely what it abandons, for the reason it states plainly: the lists get long and entries get missed.
"Take the prime that appears more often, since we want the highest common factor." The word highest attaches to the finished factor, not to the count of each prime. Each prime is capped by whichever number owns fewer of them. Taking more would produce something that is no longer a factor of both — and therefore not a common factor at all, let alone the highest one.
"The same number always supplies the ceiling." In Example 5 the 3s are capped by 750 and the 5s are capped by 225. The comparison is made prime by prime, independently.
"If the HCF is 1 the numbers have no factors in common." They share the factor 1. The chapter is careful about this: it says 1 is the only common factor and that it is also the HCF.
"The largest tile is obviously right." The book asks the reader to explain why and does not explain it. It is worth a sentence: a larger tile means fewer tiles, and fewest tiles was the condition. Present that as reasoning being supplied, not as something read off the page.
"HCF only makes sense for two numbers." The chapter says the same minimum-across-all-the-factorisations procedure covers three or more, and one exercise item asks for the HCF of three numbers at once.

Questions to check understanding

Find the HCF of two or three given numbers, showing the factorisations
List all the common factors of two numbers
Word problems of the "largest identical piece" kind: tiles, bags, ribbon lengths, rows of equal size
Given two numbers and a proposed common factor, decide whether it is the highest and justify the decision
Given a pair whose HCF is 1, say what that means about their factorisations
Explain why a non-prime splitting cannot settle whether two numbers share a factor
The closing exercise block asks this as a multiple-choice item — the largest number dividing both 306 and 36 (Part II, p.64, question 7) — and as a packing problem, the cube sizes that fill a 12 by 18 by 36 box (Part II, p.64, question 6)

Examples worth working on the board

Sameeksha's room (Part II, §3.1, p.47). Checked against the printed page. Inputs: the room is 12 ft by 16 ft; the tiles are square, all the same size, and the side must measure a whole number of feet; she wants as few of them as possible. The page carries a hand-drawn rectangle with 16 ft along the top and 12 ft down the right side. The book lists the factors of 12 as 1, 2, 3, 4, 6, 12 and of 16 as 1, 2, 4, 8, 16, so the common factors are 1, 2 and 4, and it settles on a 4 ft tile. How many tiles that takes is asked and not answered — leave the count to the explanation. So is the follow-up question of what happens if fractional side lengths are allowed.
Lekhana's rice (Part II, §3.1, p.48). Inputs: 84 kg from one farm, 108 kg from the other; every bag the same whole number of kilograms; no bag mixes the two farms; as few bags as possible. The book prints both factor lists in full and then the common factors: 1, 2, 3, 4, 6 and 12. It stops at the question of which to choose.
Jump Jackpot (Part II, §3.1, p.48). Four pairs of treasure positions: 14 and 30; 7 and 11; 30 and 50; 28 and 42. The jumper starts at 0 and takes equal strides. The closing question — whether the longest usable stride is always the HCF — is the one worth working through.
Example 1 (Part II, §3.1, p.52). 45 = 3 × 3 × 5 and 75 = 3 × 5 × 5. The page prints three matching pictures, each showing the same subpart ringed in red inside both rows: first a single 3, then a single 5, then 3 and 5 together. The common factors the book collects are 3, 5 and 3 × 5, together with 1, and the HCF it reports is 15.
Example 2 (Part II, §3.1, pp.52–53). 112 = 2 × 2 × 2 × 2 × 7 and 84 = 2 × 2 × 3 × 7. Checked against the printed page: five matching pictures run down p.53, ringing in turn 2, then 7, then 2 × 2, then 2 × 7, then 2 × 2 × 7. The HCF the book reports is 2 × 2 × 7 = 28.
Example 3 (Part II, §3.1, p.53). 96 = 2 × 2 × 2 × 2 × 2 × 3 and 275 = 5 × 5 × 11. No subpart can be ringed in both. The book's conclusion: the only common factor, and therefore the HCF, is 1.
Example 4 (Part II, §3.1, pp.53–54). 30 = 2 × 3 × 5 and 72 = 2 × 2 × 2 × 3 × 3. Checked against the printed page: on p.54 the two rows are printed with the shared 2 and the shared 3 boxed in red and joined across the rows, and the HCF reported as 2 × 3 = 6. The book's stated reason in each case is that 30 owns only one copy.
Example 5 (Part II, §3.1, p.54). 225 = 3 × 3 × 5 × 5 and 750 = 2 × 3 × 5 × 5 × 5. Checked against the printed page: the same boxed-and-joined picture appears at the right of the page, with one 3 and two 5s carried down, and the HCF reported as 3 × 5 × 5 = 75. Note which number supplies the ceiling in each case — 750 for the 3s, 225 for the 5s. The ceilings come from different numbers, and that is worth attention.
The two exercise sets (Part II, §3.1, "Figure it Out", pp.53–54). First set, common factors and HCF: 50 and 60; 140 and 275; 77 and 725; 370 and 592; 81 and 243. Second set, HCF only: 24 and 180; 42, 75 and 24; 240 and 378; 400 and 2500; 300 and 800. Give these as inputs. One warning worth passing when explaining it: 370 = 2 × 5 × 37 and 592 = 2 × 2 × 2 × 2 × 37 share the prime 37, which is easy to overlook, and an explanation that stops at the shared 2 will report a wrong answer.
The 72-and-144 question (Part II, §3.1, "Figure it Out", p.54). Inputs: 72 written as 6 × 12 and 144 written as 8 × 18. Nothing visibly matches. The book asks whether that licenses the conclusion that the two numbers share no factor beyond 1. It does not, and the reason is that neither splitting is into primes.

Figures to have open

The tiled-room diagram: a rectangle whose two side labels are adjustable, and a square tile that can be tiled across it at several side lengths so the leftover strip appears and disappears. Standard schematic — redraw, do not lift the book's hand-drawn rectangle.
Two rows of prime factors with a ringing mechanism that can highlight the same subpart in both. This is the workhorse figure of the topic and is reused in LCM: take the most occurrences of each prime.
A number line with two marked positions and a variable stride, for section 5.
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

NCERT Ganita Prakash, Class 7, Part II, printed Chapter 3 "Finding Common Ground", §3.1 "The Greatest of All", pp.47–49 — the tiling problem, the naming of HCF and GCD, the rice problem, Jump Jackpot, and the paragraph that abandons listing
Same section, "Finding the HCF of Numbers Using Prime Factorisation", Part II, pp.52–53 — Examples 1, 2 and 3
Same section, Part II, pp.53–54 — Examples 4 and 5 and the statement of the minimum rule, with the extension to more than two numbers
Same section, the two "Figure it Out" blocks, Part II, pp.53–54
Same chapter, SUMMARY, Part II, p.65, the HCF bullet and its two sub-bullets
Same chapter, closing exercises, Part II, p.64, questions 6 and 7
Backward pointer: Reading every factor of a number off its prime factorisation for the subpart argument this topic runs on two numbers instead of one
Forward pointer: LCM: take the most occurrences of each prime for the mirror-image argument that produces the LCM