HCF and LCM in word problems
Use HCF to cut or share into the largest equal parts. Use LCM to find when repeating events next happen together.
HCF or LCM? Read the clue
| Situation | Use | Why |
|---|---|---|
| Largest square tile to cover a floor exactly | HCF | The side must divide both the length and the breadth |
| Biggest equal boxes for two kinds of sweets, none left over | HCF | The box size must divide both counts |
| When two bells next ring together | LCM | The time must be a multiple of both intervals |
| Least number of laddoos that can be packed in 6s or in 8s | LCM | The number must be a multiple of both |
| Two runners on a circular track meet again at the start | LCM | Each runner must finish whole rounds |
Example 1 (HCF): stacking library books
A school library has 96 Hindi books and 120 English books. Each stack must have the same number of books and only one language. Find the largest number of books in a stack, and the number of stacks.
- We want the largest number that divides both 96 and 120, so we find the HCF.
- 96 = 2⁵ × 3
- 120 = 2³ × 3 × 5
- HCF = 2³ × 3 = 24
- Hindi stacks = 96 ÷ 24 = 4. English stacks = 120 ÷ 24 = 5.
- Total stacks = 4 + 5 = 9
24 books in each stack, and 9 stacks in all.
Example 2 (LCM): traffic lights in Chennai
At a junction in Chennai, three traffic lights change every 48 seconds, 72 seconds and 108 seconds. They change together at 8:00 a.m. When will they next change together?
- We want the first time that is a multiple of all three intervals, so we find the LCM.
- 48 = 2⁴ × 3, 72 = 2³ × 3², 108 = 2² × 3³
- LCM = 2⁴ × 3³ = 16 × 27 = 432 seconds
- 432 seconds = 7 minutes 12 seconds (7 × 60 = 420, and 432 − 420 = 12).
- Check: 432 ÷ 48 = 9, 432 ÷ 72 = 6, 432 ÷ 108 = 4, all whole numbers.
They next change together at 8:07:12 a.m., that is 7 minutes 12 seconds after 8:00 a.m.
Example 3 (LCM with a remainder)
Find the least number which leaves a remainder of 3 when divided by 12, 15 and 20.
- First find the least number divisible by 12, 15 and 20: their LCM.
- 12 = 2² × 3, 15 = 3 × 5, 20 = 2² × 5
- LCM = 2² × 3 × 5 = 60
- Add the remainder: 60 + 3 = 63
- Check: 63 = 12 × 5 + 3 = 15 × 4 + 3 = 20 × 3 + 3.
The least such number is 63.
Wedding caterers use the HCF to fill the most plates with equal sweets. Festival light strings blinking at different speeds flash together after the LCM of their timings.
3 marks: “Two milk tankers hold 850 litres and 680 litres. Find the largest container that can measure the milk of each tanker an exact number of times.” 850 = 2 × 5² × 17 and 680 = 2³ × 5 × 17. HCF = 2 × 5 × 17 = 170. The container holds 170 litres (5 times and 4 times).
3 marks: “Two buses leave a depot in Pune every 15 minutes and every 25 minutes. They leave together at 6:00 a.m. When will they next leave together?” 15 = 3 × 5 and 25 = 5². LCM = 3 × 5² = 75 minutes = 1 hour 15 minutes. They next leave together at 7:15 a.m.
Split it: HCF. Meet again: LCM.