Question

An eccentric clockmaker built three different clocks. The first clock was a five-minute clock designed with an alarm set to sound each time the hand reached the number 2. The second clock was a six-minute clock designed to sound each time the hand reached the number 3. The third clock was a seven-minute clock designed to sound each time the hand reach the number 4. The clockmaker started the clocks simultaneously on day, and each clock began to sound at its appropriate time. Was there a time when all three clocks sounded their alarms together? If so, tell when it occurred and explain why. If not, explain why not.

Answer 1

Yes, there was a time when the three alarms sounded together. It occurred 3 hours and 27 minutes after the clocks were started, and every 3 hours 30 minutes after that.

Explanation:

We assume the clocks are numbered 1–5 on the 5-minute clock, 1–6 on the 6-minute clock, and 1–7 on the 7-minute clock. Then the alarm on each clock goes off 3 minutes before the clock repeats its timing action.

The least common multiple of the clock times is 5·6·7 = 210 minutes, or 3 1/2 hours. All clocks will simultaneously be 3 minutes before their repeat at 3 minutes before this 210-minute period is up.

That is, the clocks will simultaneously alarm 207 minutes after being started, and every 210 minutes after that.

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Comment on clock face numbering

If the clock faces are numbered 1–12, so the 5-minute clock alarms 5·(2/12) minutes = 50 seconds after being started, for example, then the alarms can never sound together. The clocks will come together at least once on any/every multilple of 1 minute, but not on any/every multiple of 10 seconds.