# What is the minimum number of rotations Gear 1 requires to return to this starting position?

## Two gears are shown below in their starting position. Gear 1 has 6 teeth Gear 2 has 8 teeth As Gear 1 turns, it cause Gear 2 to turn at a different rate Gear 1 is rotated until the two gears are back to this starting position?? What is the minimum number of rotations Gear 1 requires to return to this starting position? A. 48 rotations B. 24 rotations C. 4 rotations D. 2 rotations I know the answer is C, but I chose D and I do not get how to get that number... Thanks

Nov 11, 2017

We can figure this out by finding the LCF.

#### Explanation:

gear 1 will be $S$

gear 2 will be $L$

.

$S = 6 , 12 , 18 , \textcolor{red}{24}$ -

• gear 1 turn rotations.
gear 1 moves in a rotation of 6

$L = 8 , 16 , \textcolor{red}{24}$ -

• gear 2 turn rotations
gear 2 moves in a rotation of 8

factors that make up $24$ are $6 \cdot 4$ and $8 \cdot 3$

we can remove $8 \cdot 3$ because neither gear has odd teeth and $8$ is not a factor in $S$

$6$ does not show up in $L$ so we are left with the only choice which is as you mentioned the correct answer $4$