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Four gears with 21, 49, 35, and 14 teeth mesh together. If one of them rotates, the other also rotates. How many times does each rotate before returning to their position at the start?

Correct answer:

a =  70
b =  30
c =  42
d =  105

Step-by-step explanation:

$$\begin{gathered} 21=3\cdot 7 \\ 49=7^2 \\ 35=5\cdot 7 \\ 14=2\cdot 7 \\ LCM(21,49,35,14)=2\cdot 3\cdot 5\cdot 7^2=1470 \\ n=LCM(21,49,35,14)=1470 \\ a=n/21=1470/21=70 \end{gathered}$$

$b=n/49=1470/49=30$

$c=n/35=1470/35=42$

$d=n/14=1470/14=105$

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