Real-world examples of least common multiple (LCM) problem solving

Everyday examples of least common multiple (LCM) problem solving

Let’s start with what most people really want: clear, concrete situations where LCM actually helps. Each example of LCM problem solving below walks through the setup, the math, and the interpretation.

1. Bus schedule alignment – a classic example of least common multiple (LCM) problem solving

Imagine two bus lines:

• Bus A comes every 12 minutes.
• Bus B comes every 18 minutes.

You arrive at the station and see both buses at the same time. You want to know when that will happen again. This is a textbook example of least common multiple (LCM) problem solving.

You’re looking for the smallest time (in minutes) that is a multiple of both 12 and 18.

Factor the numbers:

• 12 = 2² × 3
• 18 = 2 × 3²

To get the LCM, take the highest power of each prime:

• LCM = 2² × 3² = 4 × 9 = 36

So both buses will next arrive together in 36 minutes. In other words, every 36 minutes, the schedule lines up. In real life, transit planners use this kind of reasoning when coordinating connections between different routes.

2. Workout cycles – another example of LCM in planning

Say you’re planning your workouts:

• You do leg day every 4 days.
• You do cardio day every 6 days.

You did both on the same day this Monday. When will that happen again? This is a simple example of LCM problem solving with small numbers.

Prime factorization:

• 4 = 2²
• 6 = 2 × 3

Highest powers:

• LCM = 2² × 3 = 4 × 3 = 12

So every 12 days, leg day and cardio day will land on the same date again. If you started on Monday, count 12 days forward: that’s a Saturday of the following week. This kind of pattern planning shows up in habit tracking, training plans, and even medication schedules.

3. School timetable: combining different class rotations

Now let’s move to something a bit more realistic for students.

Suppose:

• Your science lab meets every 5 school days.
• Your art class meets every 8 school days.
• Today, you have both.

You want to know when you’ll next have both science lab and art on the same day. This is one of the best examples of least common multiple (LCM) problem solving with three numbers.

Prime factorization:

• 5 = 5
• 8 = 2³

LCM of 5 and 8 is 5 × 2³ = 5 × 8 = 40.

So every 40 school days, those two classes will line up again. Teachers and school schedulers use this kind of thinking when they design rotating block schedules so that certain activities coincide at predictable intervals.

For more on how school schedules are structured and optimized, the National Center for Education Statistics at nces.ed.gov has data and reports on instructional time and scheduling patterns.

4. Packaging and inventory – real examples from business

A small business sells markers in boxes and bundles:

• One supplier ships markers in boxes of 24.
• Another supplier ships markers in packs of 30.

The store manager wants to create identical classroom kits with no leftover markers, using only full boxes and full packs. How many markers should be in each kit if she wants the smallest possible kit size that works for both suppliers?

This is a practical example of least common multiple (LCM) problem solving in retail and inventory.

Prime factorization:

• 24 = 2³ × 3
• 30 = 2 × 3 × 5

Take the highest powers:

• LCM = 2³ × 3 × 5 = 8 × 3 × 5 = 120

So each classroom kit should have 120 markers. That way:

• From 24-marker boxes: 120 ÷ 24 = 5 boxes per kit.
• From 30-marker packs: 120 ÷ 30 = 4 packs per kit.

No markers are wasted, and everything is neatly divided. This same logic appears in manufacturing, shipping, and warehouse optimization.

5. Blinking lights and signals – timing examples include LCM

Imagine three warning lights that blink at different intervals:

• Light A blinks every 10 seconds.
• Light B blinks every 15 seconds.
• Light C blinks every 20 seconds.

They all just blinked together. When will they all blink together again?

This is a clean example of LCM problem solving with three numbers.

Prime factorization:

• 10 = 2 × 5
• 15 = 3 × 5
• 20 = 2² × 5

Highest powers of each prime:

• 2² (from 20)
• 3 (from 15)
• 5 (from any of them)

LCM = 2² × 3 × 5 = 4 × 3 × 5 = 60.

So every 60 seconds, all three lights blink together. Engineers use this type of reasoning in designing repeating signals, traffic lights, and certain digital timing circuits.

If you’re curious how timing and synchronization show up in computer science, MIT’s OpenCourseWare at ocw.mit.edu has free lecture notes on discrete mathematics and algorithms that touch on related ideas.

6. Fractions and LCM – an example of adding unlike denominators

LCM isn’t just about time and schedules. It’s also the workhorse behind adding and subtracting fractions.

Suppose you want to add:

\( \frac{5}{12} + \frac{7}{18} \)

You need a common denominator. The least common denominator is just the least common multiple of 12 and 18.

We already found above that:

• LCM(12, 18) = 36

Rewrite each fraction with denominator 36:

• \( \frac{5}{12} = \frac{5 × 3}{12 × 3} = \frac{15}{36} \)
• \( \frac{7}{18} = \frac{7 × 2}{18 × 2} = \frac{14}{36} \)

Now add:

\( \frac{15}{36} + \frac{14}{36} = \frac{29}{36} \)

That’s it. Every time a teacher says “find a common denominator,” you’re looking at another example of least common multiple (LCM) problem solving, just dressed in fraction clothing.

For more practice with fractions and LCM, Khan Academy offers free exercises and videos at khanacademy.org.

7. Event planning and repeating activities – a real example of schedule syncing

You’re organizing community events:

• A neighborhood cleanup happens every 14 days.
• A food drive happens every 21 days.

You’d like to plan a big joint event on a day when both are already scheduled, to boost turnout. You need to know when both events will land on the same day again.

Prime factorization:

• 14 = 2 × 7
• 21 = 3 × 7

Highest powers:

• 2 (from 14)
• 3 (from 21)
• 7 (from both)

LCM = 2 × 3 × 7 = 42.

So every 42 days, both events will coincide. This is a very natural example of LCM problem solving in community planning, sports leagues, and workplace shift rotations.

8. A slightly harder example of LCM with three numbers

Let’s push things a bit:

You’re designing a repeating schedule that must fit three cycles:

• Task A repeats every 9 days.
• Task B repeats every 12 days.
• Task C repeats every 15 days.

You want to know when all three tasks will fall on the same day again. This is one of the best examples of least common multiple (LCM) problem solving if you’re practicing for exams.

Prime factorizations:

• 9 = 3²
• 12 = 2² × 3
• 15 = 3 × 5

Take the highest powers of each prime across all three numbers:

• 2² (from 12)
• 3² (from 9)
• 5 (from 15)

LCM = 2² × 3² × 5 = 4 × 9 × 5 = 36 × 5 = 180.

So every 180 days, all three tasks line up on the same day. When you see a word problem involving multiple repeating cycles, that’s a strong hint that LCM is the right tool.

How to spot when an LCM example is hiding in a word problem

Many students struggle not with the calculation, but with recognizing when to use LCM. Let’s turn the examples above into a quick mental checklist.

Look for phrases like:

• “Every X days/minutes/hours…” for two or more events.
• “When will they happen at the same time again?”
• “Smallest number of items so that…”
• “No leftovers,” “no remainder,” or “split evenly into groups.”

If the problem is asking for a smallest shared repeat or a smallest shared size, you’re probably looking at another example of least common multiple (LCM) problem solving.

By contrast, if the question is about dividing something into equal parts or sharing with nothing left over, that’s often a greatest common divisor (GCD) problem instead.

The National Council of Teachers of Mathematics has helpful guidance on teaching and recognizing these structures in word problems at nctm.org.

Different ways to solve examples of least common multiple (LCM) problem solving

There isn’t just one way to find LCM. When you work through real examples, you’ll see at least three common methods:

Prime factorization method

This is the one we used in most examples:

1. Break each number into prime factors.
2. For each prime, take the highest exponent that appears.
3. Multiply those together.

This method is especially friendly for explaining why the LCM is what it is. It’s great for teaching, proofs, and deeper understanding.

Listing multiples method

For smaller numbers, you can simply list multiples.

Take 4 and 6:

• Multiples of 4: 4, 8, 12, 16, 20, 24, …
• Multiples of 6: 6, 12, 18, 24, …

The first common one you see is 12, so LCM(4, 6) = 12.

This method is intuitive, but it gets messy for larger numbers like 72 and 90.

Using GCD (greatest common divisor)

There’s a neat relationship:

\[ \text{LCM}(a, b) = \frac{|a · b|}{\text{GCD}(a, b)} \]

So if you can find the GCD (for example, using the Euclidean algorithm), you can get the LCM quickly.

Example: LCM of 24 and 30.

• GCD(24, 30) = 6.
• LCM(24, 30) = (24 × 30) ÷ 6 = 720 ÷ 6 = 120.

This is very handy in higher-level math, coding, and competitive problem solving.

Why these real examples of LCM matter beyond school

Let’s be honest: nobody is wandering around thinking, “Today I will perform least common multiple (LCM) problem solving.” But the patterns behind the best examples of LCM show up everywhere:

• Coordinating schedules: work shifts, transportation, rotating duties.
• Designing repeating events: practice sessions, meetings, maintenance cycles.
• Handling fractions: especially in science and engineering calculations.
• Synchronizing systems: computer processes, periodic signals, and more.

The more examples of least common multiple (LCM) problem solving you work through, the more comfortable you become with recognizing these patterns quickly. That’s what actually pays off later—whether you’re planning your calendar or debugging a timing issue in code.

For a deeper dive into number theory ideas that grow out of LCM and GCD, you can explore open textbooks and lecture notes from universities like UC Berkeley and MIT, many of which are indexed through oercommons.org.

FAQ: common questions about LCM with real examples

Q1. Can you give a simple example of least common multiple (LCM) for beginners?
Yes. Take 6 and 8. Their multiples are:

• 6: 6, 12, 18, 24, 30, 36, …
• 8: 8, 16, 24, 32, …

The first number that appears in both lists is 24, so 24 is the LCM. This is one of the simplest examples of LCM problem solving and a good starting point for younger students.

Q2. How do I know if a word problem is an example of LCM or GCD?
If the problem is about repeating events lining up, smallest common size, or least time until something happens together, it’s almost certainly an example of least common multiple (LCM) problem solving. If it’s about dividing something into the largest equal parts, or splitting without leftovers, it’s usually a GCD example.

Q3. Are there real examples of LCM in science and engineering?
Absolutely. In physics and engineering, LCM ideas appear when combining periodic motions or signals—for example, two waves with different periods repeating in sync after some time. In computer science, scheduling tasks that run at different intervals is another example of LCM-style thinking.

Q4. What is an example of using LCM in everyday life that doesn’t involve time?
The packaging problem above is a good one: choosing a number of items (like markers, bottles, or snacks) so that you can use full boxes from different suppliers with no leftovers. That’s an example of least common multiple (LCM) problem solving that shows up in business and home organizing.

Q5. What are the best examples to practice before a test?
Mix it up. Practice:

• Two-number time problems (like buses or alarms).
• Three-number schedule problems (like tasks repeating every 9, 12, and 15 days).
• Fraction problems where you find the least common denominator.
• Realistic word problems about packaging, grouping, or planning events.

Those give you a solid range of examples of least common multiple (LCM) problem solving that mirror what usually appears on quizzes and standardized tests.