Reading co-primality and divisibility straight off the prime factorisation

What to assume they know

• Prime factorisation, and why there is only one of them per number: producing a prime factorisation and reading off its prime factors
• Co-prime pairs, and why sharing no factor is what matters: co-primality defined by common factors
• Common factors: which jump sizes can land on a number: factors and common factors, and why the factor-list method is slow
• Multiplying a short string of small primes back together to check a factorisation

What they should be able to do

• Decide whether two numbers are co-prime by comparing their prime factorisations
• Explain why looking at primes only is enough, even though a shared factor might have been composite
• Decide whether one number divides another from the two factorisations, without dividing
• State the inclusion test in the form that counts repeats, not merely presence
• Produce a pair that shares every prime and still fails to divide, and say why
• Predict an answer before factorising, then check it — the habit the exercise set asks for
• Say why two different primes are always a co-prime pair, and what happens if the two primes are the same

Where it usually goes wrong

• "They share a prime, so one divides the other." 56 and 63 share 7 and neither divides the other. Sharing rules out co-primality; it says nothing about divisibility.
• "All the primes of the smaller number appear in the bigger one, so it divides." 42 and 12, printed on p.122 for exactly this reason. Presence is not enough; the copies have to be there too.
• "You only need to check the primes because the book says so." The book does not just say so — it raises the composite-common-factor objection itself on p.120 and answers it on p.121. Skipping that beat turns a proved method into a remembered one.
• "To test divisibility I should just divide." Sometimes faster, sometimes not, and it does not generalise: the exercise on p.122 gives two numbers only as factorisations, so there is nothing to divide unless you multiply them out first.
• "Co-prime and 'no factors in common at all' are different things." They are the same thing, because 1 divides everything and is always common. The definition says "no common factor above 1" for that reason alone.
• "Any two primes are co-prime, so 7 and 7 are co-prime." They are not. The claim needs the two primes to be different, and question 4 is a good place to make the class say so out loud.

Questions to check understanding

• Given two factorisations, say whether the pair is co-prime
• Given two numbers, factorise both and decide co-primality
• Decide whether one number divides another using factorisations only
• Explain why a pair that shares all its primes may still fail the divisibility test
• Given a pair described only by their factorisations, answer both questions without multiplying out
• Assess the claim that any two primes are co-prime, and state the condition it needs
• The appendix bound with this chapter answers the §5.4 questions on footer pages 7, 8 and 9 — the p.122 set that this topic rests on runs across footer pages 8 and 9

Examples worth working on the board

Every factorisation below is printed in the book.

• 56 and 63 (§5.4, p.120). Printed as 56 = 2 × 2 × 2 × 7 and 63 = 3 × 3 × 7. The 7 sits in both, so the pair fails. This is the same question the section opened with on p.117 and could not settle then.
• 80 and 63 (p.120). Printed as 80 = 2 × 2 × 2 × 2 × 5 and 63 = 3 × 3 × 7. Nothing is shared. The book then raises its own objection before allowing the conclusion.
• The book's challenge (p.120). Suppose the pair had a common factor that was composite — would its primes have to turn up in both factorisations? The question is printed and left for the reader; the answer follows on p.121, that no common prime means co-prime. Stage this as a genuine objection, because it is the one thing standing between the method and being a guess.
• 40 and 231 (p.121). Printed as 40 = 2 × 2 × 2 × 5 and 231 = 3 × 7 × 11. Prime factors 2 and 5 against 3, 7 and 11.
• 242 and 195 (p.121). Printed as 242 = 2 × 11 × 11 and 195 = 3 × 5 × 13. Prime factors 2 and 11 against 3, 5 and 13. Useful because 11 appears twice and is still one prime factor — the distinction from the previous topic, reused.
• The divisibility statement (p.121). The book introduces it with a plain case: 48 divided by 12 leaves nothing, so 48 is divisible by 12. It then asks how to settle such a question without long division, and answers that the divisor's factorisation has to sit inside the other's.
• 168 and 12 (p.121). Printed as 168 = 2 × 2 × 2 × 3 × 7 and 12 = 2 × 2 × 3. The book then regroups the first as 2 × 2 × 3 × 2 × 7 and reads off 12 × 14, which is the moment the abstract test becomes a concrete quotient. Show the movement of the regrouping as physically sliding the three primes of 12 together.
• 75 and 21 (pp.121–122). Printed as 75 = 3 × 5 × 5 and 21 = 3 × 7. The 7 is nowhere in 75, so the division cannot come out. Simple failure mode: a missing prime.
• 42 and 12 (p.122). Printed as 42 = 2 × 3 × 7 and 12 = 2 × 2 × 3. Every prime of 12 does appear in 42 — and 42 is still not a multiple of 12, because 12 needs two 2s and 42 supplies one. This is the most important worked example in the topic and should get the most time.
• The exercise set (p.122). Co-primality wanted for 30 and 45, 57 and 85, 121 and 1331, and 343 and 216, with a guess asked for before the factorising. Divisibility wanted for 225 by 27, 96 by 24, 343 by 17, and 999 by 99. Then a pair given only by their factorisations — one is 2 × 3 × 7, the other 3 × 7 × 11 — to be judged for both co-primality and divisibility without ever being multiplied out. Finally a character's claim that any two primes are co-prime, to be assessed.
• The claim about two primes (p.122, question 4). Worth handling precisely. Two different primes have single-entry prime lists that cannot overlap, so they are co-prime. A prime taken twice — 5 and 5 — shares 5 and is not. The book's printed wording says "any two prime numbers".

Figures to have open

• Two side-by-side prime lists that can be matched item by item, with copies shown as separate tokens rather than as exponents. Class 6 has not met index notation in this chapter, and the whole argument in section 11 depends on the copies being individually visible.
• A regrouping movement for 168, where the tokens slide into two clusters. Standard schematic; the book prints the regrouped line but not the motion.
• A "hidden composite" figure for sections 4 and 5: a composite common factor drawn over both numbers, then broken open. Standard schematic; the book poses this in words only.
• No photograph or textbook data table is needed. §5.4 prints no diagram on pp.120–122 — checked against all three pages.

Where this sits in the book

• NCERT Ganita Prakash, Class 6, Chapter 5 "Prime Time", §5.4 "Prime Factorisation", pp.120–122 — the co-primality test and the composite-factor objection (p.120), the two worked co-prime pairs and the divisibility statement with 168 and 12 (p.121), the two failing divisibility cases and the exercise set (p.122)
• The same section's opening exchange, p.117, for the question this topic closes
• Chapter Summary, p.128, for the book's printed statements of both tests
• Solutions appendix bound with this chapter file, footer pages 7–9, for §5.4