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Bell, traffic-light and tiling questions can use the same numbers yet need opposite tools. The wording, not the numbers, tells you whether to find the HCF or the LCM.
What is the difference between HCF and LCM?
The HCF (highest common factor) of two numbers is the largest number that divides both exactly. The LCM (lowest common multiple) is the smallest number that both divide exactly.
The NCERT Class 10 chapter Real Numbers finds both by prime factorisation: the HCF is the product of the smallest power of each common prime factor, and the LCM is the product of the greatest power of each prime factor involved.
The same NCERT chapter states that for any two positive integers a and b, HCF(a, b) × LCM(a, b) = a × b, and its remark after Example 4 shows, with 6, 72 and 120, that it can fail for three numbers. It is the same factorising you use in checking whether 4ⁿ can end in zero.
The quickest sense check: the HCF can never be bigger than the smallest number, and the LCM can never be smaller than the largest.
How do you know if a word problem needs HCF or LCM?
Ask one question before you calculate: should the answer be bigger than the numbers given, or smaller?
LCM words: together again, at the same time, next, simultaneously, meet again at the start, smallest number divisible by, least length each rod measures exactly.
HCF words: greatest, largest, maximum size, longest tape, equal rows or groups with nothing left over, cut into identical pieces, fewest tiles or boxes.
Watch the word "fewest": the fewest tiles means the biggest tile, so it is still an HCF question.
When will bells ring together again?
Three bells ring every 9, 12 and 15 minutes. They ring together at 8:00 am. When do they next ring together?
Factorise: 9 = 3², 12 = 2² × 3, 15 = 3 × 5.
Take the greatest power of each prime: 2² × 3² × 5 = 180.
180 minutes is 3 hours, so the bells ring together again at 11:00 am.
In Exercise 1.1 of Real Numbers, Sonia drives one round of a field in 18 minutes and Ravi in 12; they start together, so when do they meet at the start again? LCM(18, 12) = 36 minutes.
Traffic lights. Three signals change every 48, 72 and 108 seconds, and they change together at 7:00:00 am. Since 48 = 2⁴ × 3, 72 = 2³ × 3² and 108 = 2² × 3³, the LCM is 2⁴ × 3³ = 432 seconds, which is 7 minutes 12 seconds. They next change together at 7:07:12 am. Convert units before writing a clock time.
Do tile and packing problems use HCF or LCM?
Tiles. A floor is 4.5 m by 3.6 m. What is the largest square tile that covers it exactly? Convert to centimetres first: 450 and 360. Since 450 = 2 × 3² × 5² and 360 = 2³ × 3² × 5, the HCF is 2 × 3² × 5 = 90 cm, and you need 5 × 4 = 20 tiles.
Turn the question around and the tool flips. "What is the smallest square you can make from 12 cm by 18 cm tiles?" asks for a length that both sides measure exactly, so it needs LCM(12, 18) = 36 cm.
Packing. A sweet shop has 144 laddoos and 180 barfis. Each box holds one kind, every box holds the same number, and the shop wants as few boxes as possible. Since 144 = 2⁴ × 3² and 180 = 2² × 3² × 5, the HCF is 2² × 3² = 36 sweets per box, which makes 4 + 5 = 9 boxes.
What about questions with remainders?
"Smallest number that leaves remainder 3 when divided by 12, 15 and 20": find LCM(12, 15, 20) = 60, then add 3. The answer is 63.
"Largest number that divides 70 and 125, leaving remainders 5 and 8": subtract the remainders first to get 65 and 117, then find the HCF, which is 13.
Check by dividing back: 63 ÷ 12 leaves 3, and 70 ÷ 13 leaves 5. For how to set out that working on paper, see our guide on writing maths steps in board exams.
The same idea returns in time and work problems solved with the LCM method, where the LCM of the days becomes the total work.
If you keep picking the wrong tool, the Fermi Problem Library has worked problems with a step-by-step solver that explains the reasoning rather than printing an answer, so you can see why a question pointed to the LCM. To revisit factors and multiples themselves, each idea has its own lesson in the Fermi Concept Library.
