Class 10 HCF and LCM Word Problems: Which Method to Use?

Start with the meaning of the answer

HCF is the greatest positive integer that divides each given positive integer exactly. LCM is the smallest positive integer that is a multiple of each. Words such as “largest” are clues, but the quantity being asked for matters more than a keyword.

Choose by the situation

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A keyword can choose the wrong operation

Two lights flash together at time zero, then every 12 seconds and 18 seconds respectively. What is the largest time less than 100 seconds when both flash together? The word “largest” does not make this an HCF problem. Coincidences are positive multiples of LCM(12,18) = 36 seconds: 36, 72, 108, … . The required time is 72 seconds. Decide what must repeat or be divided before reacting to a single keyword.

One pair of numbers, two different questions

A club has 48 pencils and 72 erasers. For the greatest number of identical packs, the pack count must divide both totals. Since 48 = 2⁴ × 3 and 72 = 2³ × 3², the HCF is 2³ × 3 = 24. Each pack contains 2 pencils and 3 erasers. The answer is 24 packs, not 24 objects per pack.

Now imagine two signals repeat every 48 and 72 seconds and start together. The next shared time is the LCM: 2⁴ × 3² = 144 seconds. Check: 144 ÷ 48 = 3 cycles and 144 ÷ 72 = 2 cycles. The same numbers need a different operation because the situation changed.

Check before moving on

For two positive integers, HCF × LCM equals their product: 24 × 144 = 48 × 72. This verifies the arithmetic; it does not verify that you chose the right operation. Also check that the result answers the story with the correct unit.

When revising, cover the calculations and explain why “divides both” fits the pack problem while “multiple of both” fits the timing problem. That one sentence is a better method check than remembering a keyword list.

Choose what to study next

Sources and scope

This is a focused learning resource, not a complete chapter or a prediction of board questions. Use your current syllabus and the paper-specific marking scheme for assessment requirements.