Math101 · Independent Ontario learning library
Math101learn.math101.caLeast Common Multiple
The least common multiple (LCM) of positive integers is the smallest positive number that is a multiple of each.
The LCM synchronizes cycles and supplies the least common denominator for fraction operations. Prime exponents reveal why the method includes exactly enough of each factor.
Intuition and core definition
The least common multiple (LCM) of positive integers is the smallest positive number that is a multiple of each. For $6$ and $8$, common multiples begin $24,48,72,\ldots$, so the LCM is $24$. It describes the first positive point at which repeating counts or schedules coincide.
Notation, language, and conditions
$\operatorname{lcm}(a,b)$ denotes the LCM. Using prime factorizations, include every prime appearing in any input and take the largest exponent needed. For positive $a,b$, $\gcd(a,b)\operatorname{lcm}(a,b)=ab$, which is a useful check.
Why this idea matters
The least common multiple finds the first positive total or time that several integer cycles can share exactly.
A dependable method
- Prime-factor every input or list multiples when numbers are small.
- Collect each distinct prime that appears.
- For each prime, choose the greatest exponent among the inputs.
- Multiply the selected powers.
- Verify that every input divides the result and that the GCF–LCM product identity holds for two inputs.
Worked example
Representations and interpretation
Two skip-counting tracks mark multiples of each number; the first shared positive mark is the LCM. A prime-exponent table gives a symbolic representation, with the maximum in each column ensuring every input factor fits.
Reasoning about variations
If one number divides the other, such as $8$ and $40$, the larger number is already the LCM. Relatively prime numbers have LCM equal to their product because they share no prime factors.
Common mistakes
How to check your work
- Divide the proposed LCM by every input and require whole-number quotients.
- Confirm no smaller listed common multiple was skipped.
- For two positive inputs, test $\gcd\times\operatorname{lcm}=a\times b$.
Practice
- Find $\operatorname{lcm}(12,18)$.
- Two lights flash every $8$ and $12$ seconds. After how many seconds do they first flash together again?
- Find $\operatorname{lcm}(9,10)$.
Answers and brief solutions
Show answers
- $36$ $12=2^2\cdot3$ and $18=2\cdot3^2$, so use $2^2\cdot3^2=36$.
- $24$ seconds $\operatorname{lcm}(8,12)=24$.
- $90$ The inputs are relatively prime, so their LCM is their product.
Synthesis and transfer
For warning lights flashing every $12$ and $18$ seconds, the LCM predicts the first reunion after the start, and division confirms an integer number of flashes for each light.
The flashing times form the sets $12,24,36,\ldots$ and $18,36,54,\ldots$, so $36$ seconds is the first positive reunion. Prime exponents give the same result by taking every required factor at its largest occurrence: $12=2^2\cdot3$ and $18=2\cdot3^2$, producing $2^2\cdot3^2$. Dividing $36$ by each period yields whole cycle counts, three and two, which verifies the schedule interpretation. If the lights begin at different times, ordinary LCM is no longer the complete model because phase offsets matter. The common-multiple idea still helps, but one must solve for times satisfying both shifted schedules rather than assume simultaneous starts.
Related topics
Teaching and accessibility note
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
Find $\operatorname{lcm}(12,18)$.
- $12=2^2\cdot3$ and $18=2\cdot3^2$, so use $2^2\cdot3^2=36$.
End of lesson
Nice work making it this far.
Understanding grows through return visits. Save this lesson, try the practice, or continue when you are ready.
