Assume that those 3 clocks alarm at the same time when the 5-min-clock-hand reached the number 2 on its a-th turning, the 6-min-clock-hand reached the number 3 on its b-th turning, and the 7-min-clock- hand reached the number 4 on its c-th turning.
The time required to sound simultaneous alarming for each clock, T5 for the 5-min-clock, T6 for the 6-min-clock and T7 for the 7-min-clock, will be written as follows: T5=5(a-1)+2=5a-3 ··· Eq.1, T6=6(b-1)+3=6b-3 ··· Eq.2 and T7=7(c-1)+4=7c-3 ··· Eq.3 Here, a, b and c are unknown integers greater than or equal to 1.
They alarm at the same time. So, T5=T6=T7 ⇒ 5a-3=6b-3=7c-3 ⇒ 5a=6b=7c. Therfore, the least common multiple(LCM) of 5a, 6b and 7c that satisfies the equation above is 210(=5x2x3x7). Thus, a:b:c=(210/5):(210/6):(210/7)=42:35:30. Therefore, the sets of {a, b, c} are as follows: {42, 35, 30}, {84, 70, 60}, {126, 105, 90}··· {42n, 35n, 30n}, n: integers greater than or equal to 1.
Plug a=42, b=35 and c=30 into Eq.1, Eq.2 and Eq.3 respectively. T5=5x42-3=207, T6=6x35-3=207 and T7=7x30-3=207 Therefore, the clocks alarm simultaneously at 207 minute for the first time after they started at the same time from each one's origin (numbered 5, 6, 7 or 0?). Then they repeat simultaneous alarming every 210(=LCM) minutes.
CK: Plug a=84, b=70 and c=60 into Eq.1, Eq.2 and Eq.3 respectively. T5= 5x84-3=417=207+210(LCM), T6=6x70-3=417=207+210(LCM) and T7=7x60-3=417=207+210(LCM) CKD.