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📰 Question: A middle school student is building a robot with gears. One gear rotates 48 times per minute, and another rotates 72 times per minute. If both gears start aligned, after how many minutes will they align again? 📰 Solution: To find when the gears align again, we compute the least common multiple (LCM) of their rotation periods. Since they rotate at 48 and 72 rpm (rotations per minute), the time until alignment is the time it takes for each to complete a whole number of rotations such that both return to start simultaneously. This is equivalent to the LCM of the number of rotations per minute in terms of cycle time. First, find the LCM of the rotation counts over time or convert to cycle periods: The time for one rotation is $ \frac{1}{48} $ minutes and $ \frac{1}{72} $ minutes. So we find $ \mathrm{LCM}\left(\frac{1}{48}, \frac{1}{72}\right) = \frac{1}{\mathrm{GCD}(48, 72)} $. Compute $ \mathrm{GCD}(48, 72) $: 📰 Prime factorization: $ 48 = 2^4 \cdot 3 $, $ 72 = 2^3 \cdot 3^2 $, so $ \mathrm{GCD} = 2^3 \cdot 3 = 24 $. 📰 Why Everyone Is Obsessed With Malenia In Elden Ring Must Know Facts 4851360 📰 Becerro 9824195 📰 Download Vectorworks Viewer 1192070 📰 You Wont Believe What This Jurassic Park Poster Reveals About The Original 1993 Masterpiece 3160502 📰 Playgames Revealed The Ultimate Collection Thatll Keep You Addicted Forever 2192279 📰 Npi Search Pharmacist Revealed Uncover Your Local Pharmacists Npi Now 4702208 📰 Unblocked Games 77 You Never Thought Possible 1739092 📰 This Unfrosted Film Will Shock You Hidden Story Behind The Blockbuster Claims 3546264 📰 Khadiyah Lewis 9729770 📰 This Explosive High5Casino Tip Is Revolutionizing How Players Win Big Today 6164633 📰 Rockford Il Obituaries 6296151 📰 Youll Never Guess What Happens When Your Data Board Crashes 2425350 📰 Graco Stock Soared 300Heres Why Investors Are Rushing To Buy Now 354757 📰 Allianz Medical Travel Insurance 8098886 📰 Hbp In Pregnancy The Silent Threat Hidden Behind Puffy Hands And Swollen Feet 6922788