Ruiya Math Blog · No. 7 · One Thinking Move
Two rhythms meet again after one shared repeat gap.
Long lists are tempting, but they hide the idea. When two repeating schedules meet once, find how far apart their beats are and use the smallest repeat that fits both.
A Ruiya-original PSLE-style practice guide for Primary 5, Primary 6 and PSLE
The first useful observation
The first display moves in jumps of 4 days; the second moves in jumps of 5 days. We need a number of days that is a multiple of both 4 and 5.
One display repeats every 4 days and the other every 5 days. Their shared beat returns every 20 days.
The challenge
The library-display challenge
- • One library display changes on days 2, 6, 10, 14, …
- • A second display changes on days 6, 11, 16, 21, …
- • Day 6 is the first day both displays change together.
- • Find the sum of the next two days on which they change together.
This is a Ruiya-original analogue, not a past-paper question.
Why this strategy fits
Use a picture your brain already understands.
Think of two drummers. One taps every 4 beats and one taps every 5 beats. After they clap together, the next shared clap comes when both patterns have completed a whole number of beats.
Find each rhythm. 2, 6, 10, 14 shows a gap of 4. 6, 11, 16, 21 shows a gap of 5.
Find the shared repeat gap. The smallest number divisible by both 4 and 5 is 20. This is the least common multiple of the two gaps.
Jump from the known meeting day. They meet at day 6, then every 20 days: day 26 and day 46. Their sum is 26 + 46 = 72.
Check your answer
Day 26 is in the first pattern (2 + 6 × 4) and the second pattern (6 + 4 × 5). Day 46 works the same way.
Answer: The next two shared days are 26 and 46, so their sum is 72.
Traps to watch for
The usual wrong turn is often a very reasonable first thought.
Adding the gaps
4 + 5 = 9 is not a shared repeat. Both rhythms must land exactly on the same day.
Starting from day 0
The shared starting point given is day 6. Add the repeat gap from there.
Extending a very long list
A few terms help you spot the gaps, but the repeat idea is safer and faster than guessing down a page.
How an examiner can change it
Keep the thinking move, not the story.
The setting can be bus arrivals, flashing lights, school duties or medication schedules. First find the repeat gaps; then use their smallest common multiple from the stated shared starting point.
Try a fresh variation
One lift is serviced every 6 days and another every 8 days. They are both serviced today. On which day will they next be serviced together?
Do not look for the same numbers. Ask which relationship is still doing the work.
Outside the exam
A useful model helps you make a fair decision.
Maintenance calendars, transport timings and rotating duties rely on shared cycles. A repeat model helps planners know when two schedules will clash or line up again.
Try it with Ruiya
Want help choosing the first step?
Send Ruiya a hard question and ask: “What is the relationship I should notice before I calculate?”
Try 20 Free Turns