Divisibility by 6 and other numbers, checked through their factors

What to assume they know

• Prime factorisation, and reading a factorisation as a count of each prime
• LCM, and the rule that two divisors force divisibility by their LCM (The four divisibility facts you can prove, and how to use them)
• The digit tests for 9, 3 and 11 (The digit-sum test for 9, and the algebra underneath it, Why the same digit sum also settles divisibility by 3, The alternating-sum test for 11)
• The earlier tests for 2, 4, 5, 8 and 10, which the chapter revisits at Part I p.123
• Counterexamples, and what a single one settles (Always, sometimes, or never: one counterexample settles it)

What they should be able to do

• Test divisibility by 6 by testing 2 and 3, and verify the result by dividing
• Show, with one number, that testing 4 and 6 does not settle divisibility by 24
• Explain the failure in terms of prime factorisation, by counting how many copies of each prime each test guarantees
• Choose a pair of divisors whose tests together do settle a given composite, and justify the choice
• State the condition on the pair in terms of shared primes, and connect it to the LCM rule proved earlier in the chapter
• Split composite divisors met elsewhere in the chapter — 12, 15, 18, 36, 44 — into testable pieces
• Fill a divisibility table efficiently by testing only the primes and their powers, and deducing the rest

Where it usually goes wrong

• "Any two factors that multiply to the divisor will do." They must also share no prime. Four and six multiply to twenty-four and settle nothing, and the chapter's own number 12 proves it.
• "The tests for 4 and 6 must be broken, then." Both tests are fine. The split is what fails. Separating a wrong tool from a wrong plan is the useful habit here.
• "Six needs a rule of its own." It does not, and the section exists to show that. Two and three between them account for every prime in six, once each.
• "If two tests work for six, two tests work for everything." Six is the easy case because its two primes are different. Twenty-four is the honest case.
• "Divisible by four and divisible by six means divisible by twelve, so twelve is the answer, so twenty-four should be too." Twelve is the answer, and twenty-four is not; the LCM is where the chain stops. Draw the two prime inventories and count.
• "A number that passes more tests is more divisible." Passing extra tests that repeat a prime adds nothing. What matters is covering each prime to its full power exactly once.
• "There is no shortcut for a divisor like seven, so there is none." The chapter says the opposite: they exist for every divisor up to a hundred, and explaining them needs machinery from later classes.

Questions to check understanding

• Decide whether a number is a multiple of six, and justify the method as well as the answer
• Give a number showing that one named pair of tests fails to settle a composite divisor
• Choose, for a given composite divisor, a pair of divisors whose tests together settle it, and justify the choice by prime factorisation
• Explain why two divisors force divisibility by their LCM and not by their product
• Complete a divisibility table efficiently and say which entries were tested and which deduced
• Find missing digits making a number divisible by a composite such as 18 or 44
• Say what would have to be true for a pair of tests to be enough, in general

Examples worth working on the board

Inputs. Values marked "printed" are the chapter's own.

• The opening question (Part I, §5.2, p.129, printed subheadings "More on Divisibility Shortcuts" and, under it, "Divisibility Shortcuts for Other Numbers"). How to find out whether a number is a multiple of six, and whether testing its factors 2 and 3 will do the job.
• The four numbers to try it on, printed (Part I, §5.2, p.129): 38, 225, 186 and 64. The instruction is to apply the tests for 2 and 3 and then to divide by 6 and compare. The design of the set: two pass only the two-test (38 and 64), one passes only the three-test (225), and one passes both and is therefore the only multiple of 6 among them (186). No number here fails both tests — if the explanation wants that fourth case it must supply its own number, 35 for instance, and say that it is doing so.
• The failing case, printed (Part I, §5.2, p.130). Testing 24 through its factors 4 and 6 does not work, and the chapter gives the reason in one number: 12 is a multiple of 4, and a multiple of 6, and not a multiple of 24.
• The repair, printed (same page): test 24 through 3 and through 8 instead. The chapter then sets the explanation as an exercise, asking why that pair works and the other does not, and naming prime factorisation as the tool. Sections 4 to 8 are the explanation carrying out that instruction.
• The prime factorisations the explanation needs. These are working added here, not the chapter's: 24 as three twos and a three; 4 as two twos; 6 as a two and a three; 8 as three twos; 3 as itself. Set them out as columns of prime tokens so the double-counted two is visible rather than argued.
• The chapter's closing remark, printed (Part I, §5.2, p.130): shortcuts of this kind exist for every divisor up to a hundred and for some beyond it, and understanding them needs ideas that come in later classes. That sentence is section 11 — it is an honest boundary marker.
• The fill-in table (Part I, §5.2, p.129). Ten numbers down the side and nine divisors across the top. The divisors, in printed order: 2, 3, 4, 5, 6, 8, 9, 10, 11. The numbers, in printed order: 128, 990, 1586, 275, 6686, 639210, 429714, 2856, 3060 and 406839. Only the first row is filled in as a worked sample; the other nine rows are empty. The instruction asks for a quick way of doing it, and the quick way is this topic's content: test the primes and the prime powers, then read off 6 and 10 from what you already have. See the notes for a caution about the printed sample row.
• The composite divisors the rest of the chapter asks about.: 18 (Part I p.132 no. 5) as 2 with 9; 44 (Part I p.133 no. 6) as 4 with 11; 36 (Part I p.133 no. 8) as 4 with 9; 15 (Part I p.133 no. 10) as 3 with 5, with 6 in the same item as 2 with 3; 12 (Part I p.133 no. 12 (iv)) as 4 with 3. Every one of these pairs shares no prime, which is why every one of them works — a pattern worth letting the student notice before it is named.
• Backward pointer, printed (Part I, §5.1, p.121). Two divisors force divisibility by their least common multiple, justified there by prime factorisation. That is the rule this whole topic is an application of.
• The chapter prints no answers to the table, to the four-number check, or to the prime-factorisation explanation it asks for.

Figures to have open

• Prime-token columns: each divisor drawn as a stack of prime tokens, so pooling two stacks and collapsing duplicates is a physical action rather than an argument. Standard schematic. The chapter asks for this reasoning on Part I p.130 and draws nothing at all.
• The single failing number placed against three test gates, two open and one shut. Standard schematic.
• A blank version of the chapter's nine-column table, with the columns that must be tested and the columns that can be deduced marked in different weights (Part I, §5.2, p.129).
• No photograph is needed.

Where this sits in the book

• NCERT Ganita Prakash Class 8, Part I, printed Chapter 5, "Number Play", §5.2 "Checking Divisibility Quickly", printed subheadings "More on Divisibility Shortcuts" and "Divisibility Shortcuts for Other Numbers", Part I pp.129–130.
• The fill-in table sits on Part I p.129, immediately above those subheadings and immediately after the compact test for eleven. It is handed to this topic rather than to The alternating-sum test for 11 because the quick way through it is the factor-based method, not any single shortcut.
• Backward pointer inside the same chapter: the LCM rule and its prime-factorisation justification, Part I p.121; the revisited tests for 2, 4, 5, 8 and 10, Part I p.123.
• Chapter-end "Figure it Out", Part I pp.132–133, items 5, 6, 8, 10 and 12 (iv), all of which need a composite divisor split.
• The chapter's SUMMARY, Part I p.134, lists the LCM rule as one of the four divisibility properties learnt.