A medium question on divisibility and factors for P5, with a step-by-step solution, diagrams and the answer. Common in the Hua Xia Cup's P4 and P5 qualifying rounds and finals; harder ones use how many factors a number has, and square numbers.
The question
Two bus routes start from the terminus in a housing estate.
Route A leaves every 4 minutes and Route B every 22 minutes.
At 6:00 in the morning both routes send off their first bus together.
From 6:00 to 10:00 in the morning (both times included), how many times do the two routes leave together?
Show the hints (2)
💡Hint 1: After leaving together, how many minutes until they next leave together? That time must be a multiple of both 4 and 22.
💡Hint 2: How many minutes from 6:00 to 10:00? Don't forget the departure at 6:00 itself.
Worked solution
Have a go first, then read the solution step by step.
1.Multiples of 22: 22, 44; the first that is also a multiple of 4 is 44. So they leave together every 44 minutes.
2.6:00 to 10:00: (10 − 6) × 60 = 240 minutes.
3.240 ÷ 44 = 5 remainder 20: 5 more joint departures after 6:00.
4.With the one at 6:00: 5 + 1 = 6.
⚠️Common mistake: leaving out 6:00 and answering 5, or using 4 × 22 = 88 minutes as the gap.
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