Concept 6 of 6
HCF & LCM Word Problems
Today's big idea
The wording picks the tool, with largest pointing to HCF and earliest to LCM.
Sharing into equal groups asks for HCF, while events meeting again asks for LCM.
How it works
👀 What's going on?
The sums in these questions are easy once you know which sum to do. Choosing between HCF and LCM is where the whole difficulty sits.
💡 Here's the trick
Look at the signals above. Words like largest, greatest and maximum are asking for the HCF. Words like smallest, earliest and together again are asking for the LCM.
🪄 How to do it
- 1.Underline the signal words, then match them to the signals above.
- 2.Pick HCF for equal shares, or LCM for meeting again.
- 3.Work the answer out using prime factors.
✍️ Try it
A gardener has 42 roses and 56 tulips for identical bunches. How many bunches at most? Then: a red bulb blinks every 10 seconds and a blue one every 12 seconds. When do they next blink together?
Identical bunches with none left over means HCF, and HCF(42, 56) is 14, so she makes 14 bunches. Blinking together again means LCM, and LCM(10, 12) is 60, so they meet after 60 seconds.
⚠️ Watch out
You might be thinking that any question with two numbers wants the LCM. Look for the signal word instead. Cutting, sharing and packing all ask for the largest, so each of them needs the HCF.
🌟 Why this matters
Filling boxes with nothing left over is an HCF question in ordinary clothes. Working out when two chores fall on the same day is an LCM one. Naming the tool first saves you from solving the wrong problem.
Exam tip: Circle the signal word before you calculate, since the wrong tool loses the whole mark.
Watch me solve one
Quanta solves it
“Watch how I think through one of these — then you try.”
THE PROBLEM
“Two bells ring every 15 min and 20 min. They ring together at 8:00 am. When will they NEXT ring TOGETHER?”
- 1
Spot the signal-word
“Next time TOGETHER” → use LCM. 🔔
- 2
Prime factorise both intervals
15 = 3 × 5. 20 = 2² × 5.
- 3
Take every prime at the highest power
2² × 3 × 5 = 60 min
- 4
Add to start time ✓
8:00 am + 60 min = 9:00 am
Now you try one — same steps, different story 👇
See it in action
⚡ Time to play!
Largest / Greatest = HCF · Smallest / Together-again = LCM.
STORY · ROUND 1 OF 3
Ravi has 24 red pens and 40 blue pens. He packs them into identical boxes using all the pens. What is the GREATEST number of boxes?
Quick check
✅ VERIFICATION
Two bells ring at intervals of 18 minutes and 24 minutes. If they ring together at 9:00 am, at what time will they next ring TOGETHER?
⚡ Finish the activity above first to answer.
Keep going!