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Practice Worksheet

10 Real-Life LCM Word Problems with Solutions

Word problems are where Least Common Multiple actually earns its keep. Every problem below includes the full setup, the prime or GCF calculation, and what the final number means in context—plus a one-click link to open the numbers in our live calculator.

Problem #1Transit Schedules
Open (12, 18) in Calculator →

Two Airport Shuttles Departing Together

Terminal A Shuttle leaves the main curb every 12 minutes, and Terminal B Shuttle leaves every 18 minutes. If both shuttles pull away from the curb together at 7:00 AM, at what time will they next leave the curb at the exact same minute?

Step-by-Step Solution

  1. Prime factorize each interval: 12 = 2² × 3¹ and 18 = 2¹ × 3².
  2. Take the highest power of each prime: 2² × 3² = 4 × 9 = 36 minutes.
  3. Add 36 minutes to the starting time of 7:00 AM.
Final Answer: 7:36 AM (after 36 minutes — 3 trips for Shuttle A and 2 trips for Shuttle B).
Problem #2Geometry & Tiling
Open (15, 20) in Calculator →

Building the Smallest Square Mosaic from Rectangular Tiles

A ceramic studio makes rectangular tiles that measure 15 cm wide by 20 cm long. Assuming zero grout gap, what is the side length of the smallest square tabletop you can tile without cutting any tiles, and how many tiles will it take?

Step-by-Step Solution

  1. For the tabletop to be a square, its side length must be a multiple of both 15 cm and 20 cm.
  2. Using the GCF shortcut: GCF(15, 20) = 5, so LCM(15, 20) = (15 × 20) ÷ 5 = 60 cm.
  3. Along the 60 cm width you need 60 ÷ 15 = 4 tiles; along the 60 cm length you need 60 ÷ 20 = 3 tiles.
Final Answer: A 60 cm × 60 cm square, using 4 × 3 = 12 whole tiles.
Problem #3Mechanical Engineering
Open (16, 24) in Calculator →

Meshing Gear Teeth Alignment

Inside a clockwork drive, a small pinion gear with 16 teeth meshes with a larger wheel gear that has 24 teeth. A white chalk mark is drawn where two teeth touch. How many tooth clicks must pass before those exact two chalk-marked teeth meet again, and how many full rotations will each gear have made?

Step-by-Step Solution

  1. Each rotation of the small gear advances 16 teeth; each rotation of the large gear advances 24 teeth.
  2. Factor into primes: 16 = 2⁴ and 24 = 2³ × 3¹.
  3. LCM(16, 24) = 2⁴ × 3¹ = 16 × 3 = 48 tooth clicks.
Final Answer: 48 tooth clicks (3 full rotations of the 16-tooth gear and 2 full rotations of the 24-tooth gear).
Problem #4Traffic Signals
Open (40, 60, 90) in Calculator →

Three Corridor Traffic Lights Turning Green Together

Three consecutive intersections along an avenue cycle their green lights every 40 seconds, 60 seconds, and 90 seconds respectively. If all three lights turn green at the exact same instant at 12:00:00 PM, how many minutes will pass before all three turn green simultaneously again?

Step-by-Step Solution

  1. Prime factorize all three cycle times: 40 = 2³ × 5, 60 = 2² × 3 × 5, and 90 = 2 × 3² × 5.
  2. Pick the highest power of each prime: 2³ × 3² × 5¹ = 8 × 9 × 5 = 360 seconds.
  3. Convert 360 seconds into minutes: 360 ÷ 60 = 6 minutes.
Final Answer: 6 minutes later (at 12:06:00 PM).
Problem #5Bakery & Recipes
Open (6, 8, 12) in Calculator →

Matching Pastry Boxes with Zero Leftovers

A bakery sells tart shells in trays of 6, fruit fillings in jars that fill 8 tarts, and glazing packets that cover 12 tarts. What is the smallest batch of tarts you can bake so that you use up whole trays, whole jars, and whole glazing packets with nothing left over?

Step-by-Step Solution

  1. Notice that 6 already divides evenly into 12, so any multiple of 12 is automatically a multiple of 6.
  2. That reduces the problem to finding LCM(8, 12).
  3. Multiples of 12 are 12, 24, 36… and 8 divides evenly into 24 (24 ÷ 8 = 3).
Final Answer: 24 tarts (4 trays of shells, 3 jars of filling, and 2 packets of glaze).
Problem #6Daily Schedules
Open (4, 6, 8) in Calculator →

Three Recurring Reminder Timers

A nurse sets three recurring station timers: Timer A chimes every 4 hours to log room temperatures, Timer B chimes every 6 hours to check IV pumps, and Timer C chimes every 8 hours to rotate shift charts. If all three chime together at 6:00 AM on Monday, when will all three chime together next?

Step-by-Step Solution

  1. Prime factorize: 4 = 2², 6 = 2 × 3, and 8 = 2³.
  2. Highest prime powers: 2³ × 3¹ = 8 × 3 = 24 hours.
  3. 24 hours after 6:00 AM Monday is 6:00 AM Tuesday.
Final Answer: 24 hours later (6:00 AM on Tuesday).
Problem #7Track & Athletics
Open (75, 90) in Calculator →

Two Runners Lapping the Starting Line Together

Elena completes one 400-meter track lap every 75 seconds, while Marcus completes one lap every 90 seconds. They start side by side on the starting line at the gun. After how many seconds (and how many laps each) will they cross the starting line side by side again?

Step-by-Step Solution

  1. Find GCF(75, 90) = 15 seconds.
  2. Apply the two-number formula: LCM(75, 90) = (75 ÷ 15) × 90 = 5 × 90 = 450 seconds.
  3. Divide 450 by each runner's lap time: 450 ÷ 75 = 6 laps for Elena; 450 ÷ 90 = 5 laps for Marcus.
Final Answer: 450 seconds (7 minutes 30 seconds), when Elena finishes lap 6 and Marcus finishes lap 5.
Problem #8Navigation
Open (10, 15, 25) in Calculator →

Three Harbor Lighthouse Beacons Flashing Together

Across a bay, three navigation buoys flash their beacons every 10 seconds, 15 seconds, and 25 seconds. If a sailor sees all three flash at the exact same second, how long until all three flash together again?

Step-by-Step Solution

  1. Prime factorize: 10 = 2 × 5, 15 = 3 × 5, and 25 = 5².
  2. Highest prime powers: 2¹ × 3¹ × 5² = 2 × 3 × 25 = 150 seconds.
Final Answer: 150 seconds (2 minutes and 30 seconds).
Problem #9Gardening & Spacing
Open (9, 12) in Calculator →

Lining Up Fence Posts and Rose Bushes

Along a straight garden path, wooden fence posts are placed every 9 feet and climbing rose bushes are planted every 12 feet, starting with both a post and a rose bush at the 0-foot gate. How far down the path is the next spot where a rose bush sits right against a fence post?

Step-by-Step Solution

  1. Prime factorize: 9 = 3² and 12 = 2² × 3¹.
  2. Multiply the highest powers: 2² × 3² = 4 × 9 = 36 feet.
Final Answer: At the 36-foot mark (the 4th fence post and the 3rd rose bush).
Problem #10Music & Polyrhythms
Open (14, 21, 35) in Calculator →

Three Metronome Loops Resolving on Beat One

An electronic composer layers three percussion loops that are 14 beats, 21 beats, and 35 beats long. If all three loops start playing on beat 1, after how many total beats will all three loops restart simultaneously?

Step-by-Step Solution

  1. Notice all three numbers are multiples of 7: 14 = 2 × 7, 21 = 3 × 7, and 35 = 5 × 7.
  2. Highest prime powers: 2¹ × 3¹ × 5¹ × 7¹ = 210 beats.
Final Answer: After 210 beats (15 cycles of the 14-beat loop, 10 cycles of the 21-beat loop, and 6 cycles of the 35-beat loop).