Reading HCF and LCM straight off two factorisations

What to assume they know

• Why a composite number has one prime factorisation and no other — that a number's prime factorisation exists and is the only one it has
• HCF and LCM from earlier classes by listing factors, listing multiples, or common division, and what each word means
• Index notation, including that a prime absent from a number can be written with index 0
• Multiplying prime powers back into an ordinary number

What they should be able to do

• State the divisibility test in index form: one number divides another when every prime index in the first is at most the matching index in the second
• Lay two factorisations out in aligned prime columns, filling zero indices for primes that only one of them carries
• Compute an HCF by taking the lower index in each shared column, and justify each choice by the divisibility test rather than by the rule
• Compute an LCM by taking the higher index in every column, and justify it the same way
• Explain why a prime appearing in only one of the numbers contributes nothing to the HCF and everything to the LCM
• Extend the same column method to three numbers without changing anything about the reasoning
• Recognise when two numbers share no prime at all, and state the HCF and LCM immediately

Where it usually goes wrong

• "HCF means multiply the small numbers, LCM means multiply the big ones." The comparison is per prime, not per number. In 6 and 20 the HCF takes its 2 from the smaller number and the LCM takes its 2 from the larger, and the 3 and the 5 each come from whichever number has them.
• "A prime in only one of the numbers goes into the HCF as well." It cannot: the other number is not divisible by it. Writing that column as index 0 turns the question into "the lower of 1 and 0", which answers itself.
• "The LCM only uses shared primes." The reverse — it uses every prime that appears anywhere, because the result must be a multiple of both numbers.
• "If two numbers share no factor, they have no LCM." They have the friendliest possible one: their product. 17, 23 and 29 give 11339 straight off.
• "You still have to list multiples to be sure." The index test is the definition of divisibility restated, so the columns are not a shortcut past the reasoning — they are the reasoning, with the search already done.
• "HCF and LCM are just words for exam questions." The track problem is the corrective: 36 minutes is the first time two whole laps counts line up, and no other quantity answers that question.

Questions to check understanding

• Compute the HCF and the LCM for a given pair by the prime factorisation method, showing both factorisations
• Do the same for three numbers, using one table with three rows
• Given the factorisations, state the HCF and LCM without computing the numbers themselves
• Decide whether a stated number is a common multiple of two given numbers by comparing indices only
• A timing or tiling word problem that resolves to a single LCM, with the reason it is an LCM and not an HCF

Examples worth working on the board

Values marked verified are worked out here on the chapter's printed data; the chapter prints no answers to its exercises, and no answer key was consulted.

• Example 2 — 6 and 20 (§1.2, p. 4). Printed inputs: 6 = 2¹ × 3¹ and 20 = 2² × 5¹. The chapter gives HCF 2 and LCM 60, then names the two rules. Verified: in column form, 2 has indices 1 and 2, 3 has indices 1 and 0, 5 has indices 0 and 1; lower indices give 2¹ × 3⁰ × 5⁰ = 2, higher indices give 2² × 3¹ × 5¹ = 60. Build the divisor list too: the divisors of 6 are 1, 2, 3, 6 and of 20 are 1, 2, 4, 5, 10, 20, so the shared ones are 1 and 2 — the rule and the list agree, and the list is what the rule replaces.
• Example 3 — 96 and 404 (§1.2, pp. 4–5). Printed inputs: 96 = 2⁵ × 3 and 404 = 2² × 101. The chapter gives HCF 4 and reaches LCM 9696 by dividing the product by the HCF. Verified by the column method instead: 2 has indices 5 and 2, 3 has 1 and 0, 101 has 0 and 1, so the HCF is 2² = 4 and the LCM is 2⁵ × 3 × 101 = 32 × 303 = 9696. The two routes landing on the same 9696 is the section's payoff; the shortcut route itself belongs to the next topic.
• Example 4 — 6, 72 and 120 (§1.2, p. 5). Printed inputs: 6 = 2 × 3, 72 = 2³ × 3², 120 = 2³ × 3 × 5. The chapter gives HCF 6 and LCM 360. Verified: the 2 column is 1, 3, 3 and the 3 column is 1, 2, 1, while 5 is 0, 0, 1; lowest across all three gives 2 × 3 = 6, highest gives 2³ × 3² × 5 = 360. Nothing about the method changed when a third number arrived — that is the point of the section.
• Exercise 1.1 Q2 — pairs to check (§1.2, p. 5): 26 and 91; 510 and 92; 336 and 54. Verified: 26 = 2 × 13 and 91 = 7 × 13, giving HCF 13 and LCM 2 × 7 × 13 = 182. 510 = 2 × 3 × 5 × 17 and 92 = 2² × 23, giving HCF 2 and LCM 2² × 3 × 5 × 17 × 23 = 23460. 336 = 2⁴ × 3 × 7 and 54 = 2 × 3³, giving HCF 2 × 3 = 6 and LCM 2⁴ × 3³ × 7 = 3024. The second pair is the useful one when explaining it: four primes in one number, two in the other, and only 2 shared.
• Exercise 1.1 Q3 — triples (§1.2, p. 5): 12, 15 and 21; 17, 23 and 29; 8, 9 and 25. Verified: 12 = 2² × 3, 15 = 3 × 5, 21 = 3 × 7, so HCF 3 and LCM 2² × 3 × 5 × 7 = 420. The second triple is three distinct primes, so HCF 1 and LCM 17 × 23 × 29 = 11339. The third is 2³, 3², 5², sharing nothing, so HCF 1 and LCM 1800.
• Exercise 1.1 Q7 — the circular track (§1.2, pp. 5–6). Data: Sonia takes 18 minutes per lap, Ravi takes 12, both start together at the same point and travel the same way round. Verified: the two return together after LCM(18, 12) = 36 minutes, from 18 = 2 × 3² and 12 = 2² × 3 giving 2² × 3². This is the item that shows what LCM is for — a moment that must be a whole number of laps for both riders at once.

Figures to have open

• A prime-column table that can hold two or three numbers as rows and primes as headed columns, with cells for indices including zero. Standard schematic; the chapter sets its work out in running text rather than in a table, so this layout is added here and is what carries the whole topic.
• A two-set diagram of the divisors of 6 and the divisors of 20 with the overlap marked, used once in section 4 and then dropped.
• A circular track with two markers moving at different lap times, ticking to 36 minutes. Standard schematic for Exercise 1.1 Q7; the chapter describes the situation in words and prints no figure for it.

Where this sits in the book

• NCERT Mathematics, Textbook for Class X, printed Chapter 1 "Real Numbers", §1.2 "The Fundamental Theorem of Arithmetic", pp. 4–5. Example 2 and the two rules are on p. 4; Example 3 concludes and Example 4 runs on p. 5.
• Exercise 1.1 Q2, Q3 and Q7, pp. 5–6. Q7 begins on p. 5 and its final sentence runs over onto p. 6.
• The naming of the approach as the prime factorisation method, §1.2, p. 4.