LCM: take the most occurrences of each prime

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What the video covers, what to say before it, where a class usually goes wrong, and what to set afterwards. An account is free, and it opens every chapter of every book.

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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 — subparts of a factorisation are exactly its factors
HCF: take the fewest occurrences of each prime — the HCF built prime by prime from the smaller count
Class 6 Prime Time: multiples, common multiples, and the Idli-Vada game
Reading a list of multiples of a number and continuing it

What they should be able to do

Recognise a "when do two repeating things line up" problem and say why it is a common-multiple problem
State what the lowest common multiple of two numbers is
Explain why there is no largest common multiple, and why that is the reason the lowest one is the useful extreme
State what the factorisation of a multiple must contain, and check it on an example
Build a common multiple by supplying every prime that either number needs
Compute the LCM directly by taking, for each prime present, the larger of its two counts
Say why supplying extra copies of a prime still gives a common multiple, and why it stops being the lowest
Extend the same procedure to three numbers

Where it usually goes wrong

"The LCM is the two numbers multiplied together." It is for 7 and 11, and it is not for 6 and 8, where the product is 48 and the answer is 24. The product is always a common multiple — that is why the LCM can never exceed it — but the shared primes get counted twice in it. Show 6 × 8 and 24 side by side.
"Take the smaller count, as we did for the HCF." The constraint has reversed. For the HCF, a prime taken too often falls out of one of the numbers; for the LCM, a prime taken too rarely fails to hold one of them. Show the two failures next to each other.
"More copies would be safer." Extra copies still give a common multiple — the book prints two such and poses a third as a question — they just stop it being the lowest. The word lowest is the whole specification.
"There must be a largest common multiple as well as a lowest." There is not: double any common multiple and you have another one. The section asks after a largest one in those words and leaves the question open; an explanation that skips it wastes the chapter's best invitation.
"A prime has to be in both numbers to enter the LCM." It does not. 5 appears only in 360 and still has to be in the LCM of 96 and 360, because without it the result is not a multiple of 360. Caution when explaining it: the book's sentence introducing this step on p.57 describes 2, 3 and 5 as appearing in both numbers, which is not so of the 5 — the working immediately below it makes clear that what is meant is the primes appearing across the two. Do not repeat the sentence as printed; state the correct condition.
"The LCM is bigger than both numbers, always." It is at least as big as each of them, and it equals the larger one whenever the smaller divides it — which is the observation What HCF and LCM do for consecutive, even, and co-prime numbers opens on.

Questions to check understanding

Find the LCM for two or three given numbers, showing the factorisations
Word problems of the "when do these coincide again" kind: bells, buses, blinking lights, days of the week, repeating colour patterns
Given two numbers and a proposed common multiple, decide whether it is the lowest and justify the decision
Given the LCM and one number, work backwards to constraints on the other
Add fractions by taking the LCM of the denominators — the use the closing exercise puts it to, in a problem the book credits to the ancient Indian mathematician Mahaviracharya (850 C.E.), whose name is worth speaking aloud (Part II, p.64, question 13)
Decide, for two different primes, where their LCM must sit relative to the two numbers and to their product (Part II, p.64, question 10)
The repeating-colour item at the head of the closing block (Part II, p.63, question 1) is an LCM question dressed as a picture
The cowherd's cows, driven in equal numbers through 3 gates at one crossing, 5 at the next and 7 at the third, with fewer than 200 animals in all — an LCM(3, 5, 7) item the book marks as folklore mathematics from Karnataka, and a third cultural anchor for this section beside the toran and the gajak (Part II, p.63, question 5)
Two more coincidence problems in the same block: 'Fire in the Mountain', where calling 6 and calling 9 leave the group whole while calling 10 does not (Part II, p.64, question 9), and the least number that every one of 1, 2, 3, 4, 5, 6, 8, 9 and 10 divides, which the book points back to Class 6 Chapter 5 (Part II, p.64, question 12)
Two Try This items that stretch the idea past a plain LCM: the smallest number that 3, 4, 5 and 7 all divide but which leaves 10 on division by 11 — the printed item asks for the smallest, and without that word it has infinitely many answers (Part II, p.64, question 8), and the dog gaining 9 feet to the rabbit's 7 from a 150-foot head start (Part II, p.64, question 11)

Examples worth working on the board

The torans (Part II, §3.2, p.55). Checked against the printed page. Inputs: Anshu's strips are 6 cm long, Guna's are 8 cm; strips are laid end to end; both hangings have to finish at the same length. The book prints the two lists it builds — 6, 12, 18, 24, 30, 36, 42, 48, 54 and 8, 16, 24, 32, 40, 48, 56, 64, 72 — notes 24 and 48 as two of the agreements, and settles on 24 cm. The page carries a colour illustration of two children on the floor laying pink and cream strips end to end.
The open question on the same page (Part II, §3.2, p.55). Whether a largest common multiple exists is asked and left unanswered. It is the best question in the section.
The gajak (Part II, §3.2, pp.55–56). Checked against the printed page; the page carries a photograph of a tray of the sweet with a caption naming its ingredients. Inputs: the shop gives it free every Monday, so every 7 days; Kabamai comes every 10 days; today is a Monday and she has just had some. The book prints multiples of 7 up to 77 and multiples of 10 up to 70, and reports 70 days.
Idli-Vada (Part II, §3.2, p.56). Four pairs: 4 and 6; 7 and 11; 14 and 30; 15 and 55. The closing question — whether the first double call is always the LCM — is the one to show. The book notes that the last two pairs are the ones where listing became tedious.
36 and 648 (Part II, §3.2, pp.56–57). Inputs as the book gives them: 36 = 2 × 2 × 3 × 3, and 648 written as 36 × 18, then as (2 × 2 × 3 × 3) × (2 × 3 × 3). The observation to draw out is that 648's factorisation holds 36's entire factorisation with some primes to spare.
Example 6 (Part II, §3.2, p.57). 14 = 2 × 7 and 35 = 5 × 7. The book prints two common multiples that are not the lowest — 2 × 7 × 5 × 7 × 3 and 2 × 2 × 5 × 7 × 7 × 11 — then asks whether 2 × 3 × 5 × 7 is one too, and then reports the lowest as 2 × 5 × 7 = 70, with the reason that stripping anything out of it breaks one of the two containments. The answer to that question is yes. 2 × 3 × 5 × 7 = 210 holds 2 × 7 and it holds 5 × 7, so 210 is a common multiple — 210 = 14 × 15 = 35 × 6 — and it belongs with the other two in the "too big" pile, not against them. An explanation that reads the unanswered question as a trap will contradict the page. None of the four printed products is a failure case: take the 5 out and 2 × 7 = 14 no longer holds 35; take the 2 out and 5 × 7 = 35 no longer holds 14.
Example 7 (Part II, §3.2, pp.57–58). 96 = 2 × 2 × 2 × 2 × 2 × 3 and 360 = 2 × 2 × 2 × 3 × 3 × 5. The book walks the three primes in turn: five 2s against three, one 3 against two, no 5 against one. It reports 2 × 2 × 2 × 2 × 2 × 3 × 3 × 5 = 1440.
The exercise set (Part II, §3.2, "Figure it Out", p.58). LCMs wanted for 30 and 72; 36 and 54; 105, 195 and 65; 222 and 370. Give these as inputs. The three-number item is the one that tests whether the rule was understood as a rule rather than as a two-column trick.

Figures to have open

Two strip-hangings of adjustable unit length that grow until their totals agree. Standard schematic — redraw; do not reproduce the book's illustration.
Two vertical ladders of multiples that can keep scrolling, with matching rungs highlightable. Reused for sections 2, 3 and 7.
A day axis with two independent repeating tick patterns, for section 4.
Two rows of prime factors with a containment highlight, so one row can be shown sitting inside another. This is the LCM counterpart of the ringing figure in HCF: take the fewest occurrences of each prime.
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", §3.2 "Least, but not Last!", pp.55–56 — the toran problem, the open question about a largest common multiple, the gajak problem, the naming of the LCM, and Idli-Vada
Same section, "Finding LCM through Prime Factorisation", Part II, pp.56–58 — 36 and 648, Example 6, Example 7, and the statement of the maximum rule
Same section, "Figure it Out", Part II, p.58
Same chapter, SUMMARY, Part II, p.65, the LCM bullet and its two sub-bullets
Same chapter, closing exercises, Part II, pp.63–64, questions 1, 5, 8, 9, 10, 11, 12 and 13 — the repeating stars, the Karnataka cowherd, the two Try This items, 'Fire in the Mountain', the two-primes multiple choice, the smallest multiple of the small numbers, and Mahaviracharya's fractions
Backward pointer: HCF: take the fewest occurrences of each prime, whose argument this one mirrors
Forward pointer: Conjecture and generalisation: what mathematicians mean by those words, where the HCF and the LCM are tied together by one product relation