English:Least Common Multiples
Least Common Multiples
Introduction
Welcome to Least Common Multiples. In this course, you will learn how to find the least common multiple, often shortened to LCM. This skill helps you notice number patterns, compare repeating events, and work with fractions.
A multiple of a whole number is the result you get when you multiply that number by a whole number. For example, 4, 8, 12, 16, and 20 are positive multiples of 4. A common multiple belongs to the multiple lists of two or more numbers. The least common multiple is the smallest positive common multiple.
The image above connects factors and multiples. Factors are numbers that multiply to make a product. Multiples are products you get by multiplying a number by whole numbers.
By the end of this aiMOOC, you should be able to:
- Recognize multiples: Build and compare lists of multiples.
- Find common multiples: Identify numbers shared by two or more multiple lists.
- Find the LCM: Use more than one method and explain why your answer is correct.
- Solve real-world problems: Use LCM when events repeat on different schedules.
- Connect LCM to fractions: See how common denominators are related to common multiples.
The video gives a grade-level introduction to least common multiples. While you watch, pause before each answer and try the next step yourself.
Building the Idea
Multiples Are Repeating Number Patterns
You can make multiples by skip-counting. To list positive multiples of 3, start with 3 and keep adding 3:
3, 6, 9, 12, 15, 18, 21, 24, ...
To list positive multiples of 4, start with 4 and keep adding 4:
4, 8, 12, 16, 20, 24, 28, 32, ...
Notice that 12 and 24 appear in both lists. They are common multiples of 3 and 4.
From Common Multiple to Least Common Multiple
A pair of numbers usually has many positive common multiples. For 3 and 4, some common multiples are 12, 24, 36, 48, and 60. The smallest positive one is 12, so the LCM of 3 and 4 is 12.
The word least matters. If you find a common multiple, keep checking whether there is a smaller positive common multiple.
Three Important Questions
When you solve an LCM problem, ask yourself:
- Is the number a multiple of the first number?: Check whether it can be divided by the first number with no remainder.
- Is it also a multiple of the second number?: A common multiple must work for both numbers.
- Is it the smallest positive common multiple?: If a smaller common multiple exists, you have not reached the LCM yet.
Method One: List the Multiples
Listing multiples is often the easiest method for small numbers.
Example: Find the LCM of 4 and 6.
Multiples of 4: 4, 8, 12, 16, 20, 24, ...
Multiples of 6: 6, 12, 18, 24, ...
The first shared positive number is 12. Therefore, LCM of 4 and 6 = 12.
A useful stopping rule is simple: as soon as you reach the first number that appears in every list, you have found the LCM.
After watching, try the same basic method for 5 and 8. You should find 40 as the first positive number in both lists.
Try It Mentally
For each pair, say a few multiples of the larger number and check whether each one is also a multiple of the smaller number.
- LCM of 2 and 5: Start with 5, then 10. The first shared multiple is 10.
- LCM of 3 and 6: The larger number, 6, is already a multiple of 3, so the LCM is 6.
- LCM of 6 and 9: Check 9, 18, 27, and so on. The first shared multiple is 18.
- LCM of 7 and 8: Because 7 and 8 have no common factor greater than 1, their LCM is 56.
Method Two: Use a Multiplication Chart
A multiplication chart can help you spot multiples quickly. Choose one number and follow its row or column. Then do the same for the other number. Look for the first product both number patterns share.
Example: Find the LCM of 5 and 6.
The positive multiples of 5 begin 5, 10, 15, 20, 25, 30. The positive multiples of 6 begin 6, 12, 18, 24, 30. The first shared product is 30, so the LCM is 30.
This method is especially useful when you know your multiplication facts but do not want to write long lists.
Method Three: Prime Factorization
For larger numbers, prime factorization can be efficient. A prime number has exactly two positive factors: 1 and itself. A factor tree breaks a number into prime factors.
The factor tree for 24 shows that 24 can be built from prime factors. One prime factorization is:
24 = 2 × 2 × 2 × 3
Now find the LCM of 12 and 18.
12 = 2 × 2 × 3
18 = 2 × 3 × 3
To build a number that both 12 and 18 divide into evenly, take enough copies of every prime factor to cover both factorizations. You need two 2s and two 3s:
LCM = 2 × 2 × 3 × 3 = 36
For Grades 5–6, use this method as a challenge or as another way to check your answer. If listing multiples is quicker and clearer, that method is completely valid.
A Visual Prime-Factor View
A prime-factor diagram can show shared prime factors and factors that belong to only one number. To build an LCM, include enough prime factors to make every starting number divide the result evenly.
Useful Shortcuts and Checks
When One Number Is Already a Multiple of the Other
If one number is a multiple of the other, the larger number is the LCM.
Example: 6 is a multiple of 3, so the LCM of 3 and 6 is 6.
Example: 20 is a multiple of 5, so the LCM of 5 and 20 is 20.
When Two Numbers Share No Factor Greater Than One
If the only positive factor two numbers share is 1, the numbers are called coprime. For two positive coprime numbers, the LCM is their product.
Example: 5 and 8 are coprime, so their LCM is 5 × 8 = 40.
Do not use this shortcut until you are sure the numbers have no common factor greater than 1.
Quick Answer Checks
Your LCM must pass all of these checks:
- Divisibility check: The LCM must be divisible by every starting number.
- Size check: The LCM cannot be smaller than the largest starting number.
- Least check: There must not be a smaller positive number divisible by all the starting numbers.
LCM in Real-Life Situations
LCM is useful when things happen again and again at different intervals.
Repeating Events
Suppose one classroom timer beeps every 4 minutes and another beeps every 6 minutes. If they beep together now, when will they next beep together?
Multiples of 4 minutes: 4, 8, 12, 16, ...
Multiples of 6 minutes: 6, 12, 18, ...
The first shared time is 12 minutes. The timers will next beep together after 12 minutes.
Schedules and Cycles
LCM can model repeating bus arrivals, flashing lights, exercise stations, music rhythms, machine cycles, or rotating classroom jobs. The key clue is that two or more repeating patterns must meet again.
Before calculating, ask: Am I looking for the first time repeated events happen together? If yes, LCM is often the right tool.
This video connects LCM and greatest common factor to word problems. Focus on how the situation tells you which idea to use.
LCM and Fractions
LCM also appears when you work with fractions. To add or subtract fractions with different denominators, you need a common denominator. The least common multiple of the denominators gives the least common denominator.
Example: For one-half and one-third, the denominators are 2 and 3. Their LCM is 6, so sixths are a useful common unit.
1/2 = 3/6
1/3 = 2/6
Now the fractions have the same denominator and can be compared or combined more easily.
Equivalent fractions show that the same amount can be named with different denominators. LCM helps you choose a shared denominator without making it larger than necessary.
LCM or Greatest Common Factor?
LCM and greatest common factor are related, but they answer different questions.
| If the problem asks... | Think about... | Example idea |
|---|---|---|
| When will repeating events happen together again? | Least common multiple | Lights flashing every 4 seconds and 6 seconds |
| What is the smallest shared multiple? | Least common multiple | Comparing lists of multiples |
| What is the largest equal group size? | Greatest common factor | Splitting objects into identical groups |
| What is the greatest factor shared by two numbers? | Greatest common factor | Comparing factor lists |
Worked Examples
Example A: LCM of 8 and 12
Multiples of 8: 8, 16, 24, 32, 40, 48, ...
Multiples of 12: 12, 24, 36, 48, ...
The first common multiple is 24, so LCM of 8 and 12 = 24.
Example B: LCM of 4, 6, and 10
Multiples of 4 include 20, 40, and 60.
Multiples of 6 include 30 and 60.
Multiples of 10 include 20, 30, 40, 50, and 60.
The first positive number shared by all three lists is 60, so LCM of 4, 6, and 10 = 60.
Example C: Explain an Error
A learner says, “The LCM of 6 and 8 is 48 because 48 is a multiple of both numbers.”
The statement has one correct part: 48 is a common multiple. But it is not the least common multiple. Since 24 is divisible by both 6 and 8, and no smaller positive common multiple works, LCM of 6 and 8 = 24.
This is why the word least must always be checked.
Interactive Tasks
Quiz: Test Your Knowledge
What is the least common multiple of 4 and 6? (12) (!8) (!18) (!24)
Which number is the least common multiple of 3 and 5? (15) (!8) (!10) (!30)
Which statement best describes a multiple? (A result of multiplying a number by a whole number) (!A number that always has exactly two factors) (!A number smaller than every factor) (!A remainder left after division)
What is the least common multiple of 6 and 9? (18) (!15) (!27) (!54)
If one bell rings every 4 minutes and another every 10 minutes, after how many minutes will they next ring together? (20 minutes) (!14 minutes) (!24 minutes) (!40 minutes)
What is the least common multiple of 5 and 20? (20) (!5) (!25) (!100)
Which method can help you find an LCM by writing repeated products? (Listing multiples) (!Subtracting the numbers) (!Rounding the numbers) (!Measuring angles)
Which number is a common multiple of 8 and 12? (24) (!16) (!20) (!30)
Why is 48 not the least common multiple of 6 and 8? (A smaller common multiple is 24) (!48 is not divisible by 6) (!48 is not divisible by 8) (!The LCM must be smaller than both numbers)
What is the least common multiple of 2, 3, and 4? (12) (!6) (!8) (!24)
Memory Game
| Multiple | A product made by multiplying a number by a whole number |
| Common multiple | A positive number that appears in the multiple lists of two or more numbers |
| Least common multiple | The smallest positive common multiple |
| Prime factor | A factor that is also a prime number |
| Factor tree | A diagram that breaks a number into factors |
| Common denominator | A denominator shared by two or more fractions |
Drag and Drop
| Match the correct terms. | Topic |
|---|---|
| Listing multiples | Writing repeated products until the first shared value appears |
| Common multiple | A value that belongs to every multiple list being compared |
| Prime factorization | Breaking a whole number into a product of prime numbers |
| Repeating event | A situation where cycles can meet again after fixed intervals |
| Least common denominator | The smallest shared denominator useful for two fractions |
...
Crossword Puzzle
| Multiple | What do you call a number produced by multiplying a given number by a whole number? |
| Common | Which word describes something shared by two or more number lists? |
| Smallest | Which word describes the positive common multiple chosen for an LCM? |
| Prime | What kind of number has exactly two positive factors? |
| Factor | What do you call a number that divides another whole number evenly? |
| Fraction | What kind of number can use an LCM to find a least common denominator? |
LearningApps
Cloze Text
Open-Ended Tasks
Easy
- Skip-count poster: Create a poster showing the first ten positive multiples of two numbers from 2 to 9, circle the common multiples, and label the LCM.
- Common-multiple hunt: Choose three pairs of small numbers, list their multiples, and explain how you know which shared number is the least one.
- Multiplication-chart detective: Use a multiplication chart to find five LCMs, then write one sentence describing a pattern you notice.
- Mini LCM explainer: Write a short explanation for a younger learner that uses the words multiple, common, and least correctly.
Standard
- Bus schedule model: Invent two buses that arrive at different regular intervals, calculate when they will next arrive together, and show your reasoning with a timeline or multiple lists.
- Fraction bridge: Choose two simple fractions with different denominators, use the LCM of the denominators to rename them with a common denominator, and explain why the values stay equal.
- Factor-tree gallery: Draw factor trees for two composite numbers, use the prime factors to find their LCM, and check the answer by division.
- Interview a scheduler: Interview a teacher, coach, family member, or worker about a repeating schedule and describe one situation where LCM thinking could be useful.
Advanced
- Compare LCM methods: Solve the same four LCM problems by listing multiples and by prime factorization, then compare which method is clearer or faster for each problem.
- Three-cycle experiment: Use three repeating actions with different intervals, predict when they will happen together, test the pattern physically or with counters, and compare the result with your calculated LCM.
- Design an LCM board game: Create a playable board or card game in which correct LCM reasoning moves players forward, and include rules plus an answer key.
- Create an LCM tutorial video: Plan and record a short teaching video with one definition, one worked example, one real-life problem, and one common mistake to avoid.
Learning Assessment
- Assessment: Explain and justify: Find the LCM of 8 and 12 in two different ways and explain why both methods must produce the same answer.
- Assessment: Error analysis: A learner claims that the LCM of 4 and 10 is 40; decide whether the claim is correct and justify your answer by finding the least positive common multiple.
- Assessment: Transfer to schedules: Two activities repeat every 6 days and every 8 days; determine when they next happen together and explain why this is an LCM problem.
- Assessment: Method choice: Compare listing multiples with prime factorization for finding the LCM of 12 and 18, and explain which method you would choose for this pair.
- Assessment: Fractions connection: Explain how the LCM of 6 and 8 can help when working with fractions whose denominators are 6 and 8, and show one valid pair of equivalent fractions.
Evidence of Learning
Strong evidence of learning shows not only that you can calculate an LCM, but also that you understand why the method works and when the idea is useful.
| Evidence | What successful learning can look like |
|---|---|
| Knowledge | You accurately use the words multiple, common multiple, least common multiple, factor, and prime factor. |
| Skill | You find LCMs by listing multiples, using known multiplication facts, and, when appropriate, using prime factorization. |
| Reasoning | You check divisibility and explain why no smaller positive common multiple works. |
| Product | You create a correct model, poster, schedule, game, written explanation, or video that teaches or applies LCM. |
| Transfer | You recognize LCM in a new repeating-event or fraction situation and choose a suitable method without being told which one to use. |
OERs on the Topic
The English Wikipedia article below can be used for extension reading. For Grades 5–6, focus first on the overview, examples, and simple applications. Some later sections use more advanced mathematics.
Linked Learning Areas
Least common multiples connect multiplication, division, factors, prime numbers, fractions, and repeating patterns. These links can help you move from one idea to another.