When will all clocks alarm sound simultaneously?

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 6 minute clock designed to sound each time the hand reached the number 3. The third clock was a 7 minute clock designed to sound each time the hand reaced the number 4. The clockmaskter started the clocks simultaneously one 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

1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5    

1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5    

1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6    

When the three numbers 2, 3 and 4 are in sequence, they will sound their alarms together. The first 27 minutes the 5 minute and 6 minute clock go off together, but the 7 minute clock is on 6.  25 minutes later the same two clocks ring, but the 7 minute clock is on 1. 25 minutes later the same two clocks ring, but the 7 minute clock is on 3, then 5, then 7, then 2 and finally 4!!!  It takes 27 + 6X25 = 177 minutes or 2 hours and 57 minutes for all 3 alarms to go off.