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how can you solve for x - ECD Germany

📅 May 4, 2026 👤 scraface

May 04, 2026

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📌 5Question: A virtual reality game designer is creating a puzzle where players must align two rotating gears with 24 and 36 teeth respectively. What is the smallest number of full rotations each gear must make before both gears align in their starting position?

📌 Solution: To find when both gears align, we need the least common multiple (LCM) of their number of teeth, 24 and 36. First, factor both numbers:

📌 $24 = 2^3 \cdot 3$,

📌 $36 = 2^2 \cdot 3^2$.

📌 The LCM is found by taking the highest powers of all primes:

📌 $ ext{LCM} = 2^3 \cdot 3^2 = 8 \cdot 9 = 72$.

📌 Thus, the smallest number of teeth rotations needed for alignment is 72. For the first gear (24 teeth), this corresponds to $72 \div 24 = 3$ rotations, and for the second gear, $72 \div 36 = 2$ rotations.

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