# Clock's gears

In the clock machine, three gears fit together. The largest has 168 teeth, the middle 90 teeth, and the smallest 48 teeth. The middle wheel turns around its axis in 90 seconds. How many times during the day do all the gears meet in the starting position?

Correct result:

n =  17

#### Solution:

$168=2^3 \cdot 3 \cdot 7 \ \\ 90=2 \cdot 3^2 \cdot 5 \ \\ 48=2^4 \cdot 3 \ \\ LCM(168, 90, 48)=2^4 \cdot 3^2 \cdot 5 \cdot 7=5040 \ \\ \ \\ z=LCM(168,90,48)=5040 \ \\ \ \\ v=90/90=1 \ \text{z/s} \ \\ \ \\ n_{1}=\dfrac{ 24 \cdot \ 3600 }{ z / v }=\dfrac{ 24 \cdot \ 3600 }{ 5040 / 1 } \doteq \dfrac{ 120 }{ 7 } \doteq 17.1429 \ \\ \ \\ n=\lfloor n_{1} \rfloor=\lfloor 17.1429 \rfloor=17$

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