LCM word problems are among the most essential application-based questions in Class 7 and Class 8 mathematics. They evaluate your ability to apply the Least Common Multiple to real-world scenarios such as bells ringing together, transport departures, gear teeth realignment, circular track laps, and tile dimensions. This guide provides 13 fully solved problems with diagrams and step-by-step solutions.
An LCM word problem is an applied mathematical scenario where you determine the smallest time period, distance, or quantity at which multiple repeating cycles synchronize.
| Real-Life Scenario | What Is Asked? | How LCM Solves It |
|---|---|---|
| Bells ringing at fixed intervals | When will all bells chime together next? | LCM of interval periods |
| Buses departing depot routes | When will all route buses depart at the same minute? | LCM of route departure minutes |
| Interlocking gear wheels | After how many teeth/rotations do marks realign? | LCM of teeth counts |
| Athletes running around an oval track | When will all runners meet at the start line? | LCM of lap completion times |
| Division with a common remainder | Smallest number leaving remainder r | LCM of divisors + remainder r |
Recognizing the problem type immediately saves precious exam time. Look for these core clues:
"Together", "at the same time", "minimum", "least", "smallest", "repeat", "coincide", "realign", "meet again". The answer is larger than or equal to the individual inputs.
"Maximum", "greatest", "largest", "longest tape", "divide equally", "equal containers". The answer is smaller than or equal to the individual inputs.
Follow this structured 4-step framework in your exams for full marks:
List the numbers given. Ensure all intervals share identical units (e.g., convert minutes to seconds if needed).
Find the LCM using prime factorization or the common division ladder method.
Convert the resulting LCM into the units requested by the question (e.g., seconds into minutes & seconds).
Add the interval to the starting clock time (if given) and state the final answer with units.
Bells ringing at periodic intervals represent the quintessential LCM word problem in school curricula.
Timeline tracks for three church bells tolling at intervals of 6, 8, and 12 minutes starting at 8:00 AM. Notice how Bell 1 rings 4 times, Bell 2 rings 3 times, and Bell 3 rings 2 times before they all toll together at 24 minutes (8:24 AM).
Question: Three bells ring at intervals of 6, 8, and 12 minutes respectively. If they toll together at 8:00 AM, at what time will they toll together next?
Question: Traffic lights at three road junctions change after every 48, 72, and 108 seconds respectively. If they change simultaneously at 7:00:00 AM, at what time will they next change together?
Question: Two alarm clocks chime at intervals of 50 minutes and 60 minutes. If both chime together at 12:00 noon, when will they next chime together?
Gears and wheels have integer counts of teeth or circumferences. The LCM determines when marked points return to their exact meshing alignment.
Gear A has 12 teeth and Gear B has 18 teeth. Marked teeth start meshed at 0. After Gear A completes 3 full turns (36 teeth) and Gear B completes 2 full turns (36 teeth), their marked teeth meet again at LCM(12, 18) = 36.
Question: Two gears have 12 teeth and 18 teeth respectively. If they start meshing at a marked tooth, after how many rotations of the smaller gear will the marked teeth mesh together again?
Question: Two bicycle wheels have circumferences of 24 cm and 36 cm respectively. What is the minimum distance the bicycle must travel so that both wheels complete an exact whole number of revolutions?
When athletes run on a closed track at constant speeds, their arrival at the starting line repeats in cycles of their individual lap times.
Three runners start together at the start line. Runner A takes 2 min per lap, Runner B takes 3 min, and Runner C takes 4 min. All three coincide at the start line together after LCM(2, 3, 4) = 12 minutes.
Question: Three athletes run around a circular track completing one lap in 2, 3, and 4 minutes respectively. If they start together from the starting point at 6:00 AM, after how many minutes will they all meet at the starting point again?
Fleet dispatchers schedule public buses and trains in repeating headways.
Question: Three inter-city buses leave a depot at intervals of 15 minutes, 20 minutes, and 30 minutes respectively. If they all depart together at 6:00 AM, at what time will they next depart simultaneously?
Question: Two express trains leave a railway junction at intervals of 40 minutes and 60 minutes. If both depart together at 9:00 AM, at what time will they next depart together?
Understanding the distinction between tiling problems that require HCF versus LCM is a classic exam distinction.
Question: Find the size of the largest square tile that can pave a rectangular courtyard of length 180 cm and width 120 cm without cutting any tiles.
Question: What is the side length of the smallest square floor that can be completely paved using rectangular tiles of size 15 cm by 20 cm?
Advanced competition problems often combine LCM with a fixed remainder.
Question: Determine the smallest number which when divided by 12, 15, and 18 leaves a remainder of 5 in each case.
Question: A shopkeeper sells sweets in packets of 15, 20, and 25 pieces. If a school wishes to buy equal counts of all three sweet varieties without opening any packet, what is the minimum count of each sweet type they must purchase?
Avoid these 5 common exam traps when tackling LCM word problems:
| ❌ Mistake | Why It Is Incorrect | ✅ Correct Procedure |
|---|---|---|
| Confusing LCM with HCF | LCM is for synchronization (larger); HCF is for partition (smaller) | If events repeat into future → LCM; if cutting or grouping → HCF |
| Forgetting to convert units | Intervals in seconds added directly to minutes without division by 60 | Convert seconds to minutes (e.g. 432 s = 7 min 12 s) before adding |
| Multiplying numbers instead of LCM | For 6 and 8, using 6 × 8 = 48 rather than LCM = 24 | Use prime factors: 23 × 3 = 24 |
| Subtracting the remainder instead of adding | Subtracting remainder 5 gives 175, which leaves no remainder when divided | Required number = LCM + Remainder |
| Missing the starting time | Writing answer as "24 minutes" instead of "8:24 AM" | Add the interval to the base clock time stated in the question |
Solve these practice problems to test your mastery:
Quick answers to common questions about Class 7/8 LCM word problems.
LCM word problems are real-life scenarios where you must find the minimum time, distance, or count where repeating events align or occur together.
LCM problems involve repeating cycles meeting simultaneously ("together", "minimum time", "least distance"). HCF problems involve dividing quantities into equal groups ("maximum size", "greatest length", "largest container").
Because each bell chimes at multiples of its period. All bells toll together at a common multiple. The very first time they do so is at the Least Common Multiple (LCM).
When a number leaves a remainder r when divided by several numbers, find the LCM of those numbers and add r: Required number = LCM + r.
Key definitions for school and Olympiad exams.