Finding the Least Common Multiple of 24 and 36
So you're staring at two numbers — 24 and 36 — and someone asks for their least common multiple. Which means what does that even mean? Turns out, it's not as mysterious as it sounds. The LCM is the smallest number that both 24 and 36 divide into evenly, with no remainder. It’s the kind of thing you might have learned in middle school and then promptly forgot, until now.
Here's the thing — knowing how to find the LCM of 24 and 36 isn't just busywork. It shows up in real situations, whether you're adding fractions, planning schedules, or trying to figure out when two repeating events line up. Let's break it down.
Most people don't realize how important this is.
What Is the Least Common Multiple?
The least common multiple (LCM) of two numbers is the smallest positive integer that is a multiple of both numbers. In simpler terms, it's the first number that appears in both lists of multiples.
For 24 and 36, that means you're looking for the smallest number you can divide by both 24 and 36 without getting a fraction or decimal. Spoiler: it's not 72. Not right away, anyway.
Why Does This Matter?
Honestly, this is the part most guides get wrong. They jump straight into the math without explaining why you'd ever care. But here's why the LCM matters:
- When adding or subtracting fractions, you need a common denominator — often the LCM of the denominators.
- If you're scheduling recurring events (like shifts, maintenance cycles, or workout routines), the LCM tells you when they'll align again.
- In music, the LCM helps determine when rhythmic patterns repeat.
If you've ever wondered when two things that happen on different schedules will sync up again, you've unknowingly been looking for an LCM.
How to Find the LCM of 24 and 36
There’s more than one way to skin this cat. Here are the three most reliable methods.
Method 1: Listing Multiples
This is the most straightforward approach, especially for smaller numbers. You list out the multiples of each number until you find a match.
Multiples of 24: 24, 48, 72, 96, 120...
Multiples of 36: 36, 72, 108, 144...
Boom. The first number that shows up in both lists is 72. That's your LCM.
But here's the catch — this method gets clunky fast with bigger numbers. You don't want to be listing multiples of 144 and 180. So let's look at better options.
Method 2: Prime Factorization
This is the method most people learn in school, and for good reason — it's reliable and works every time. Here's how it goes:
- Break each number down into its prime factors.
- For each prime number that appears, take the highest power of that prime from either factorization.
- Multiply those together.
Let's do it with 24 and 36:
24:
- 24 ÷ 2 = 12
- 12 ÷ 2 = 6
- 6 ÷ 2 = 3
- 3 ÷ 3 = 1
So the prime factorization of 24 is 2³ × 3¹.
36:
- 36 ÷ 2 = 18
- 18 ÷ 2 = 9
- 9 ÷ 3 = 3
- 3 ÷ 3 = 1
So the prime factorization of 36 is 2² × 3².
Now, take the highest power of each prime:
- For 2: the highest power is 2³ (from 24)
- For 3: the highest power is 3² (from 36)
Multiply them: 2³ × 3² = 8 × 9 = 72
The LCM is 72.
Method 3: Using the Greatest Common Divisor (GCD)
This one's a shortcut if you already know — or can easily find — the GCD. The formula is:
Want to learn more? We recommend what can i do with a chemistry degree and which chemical powder separate hydrogen from water for further reading.
LCM(a, b) = (a × b) ÷ GCD(a, b)
First, find the GCD of 24 and 36. The factors of 24 are 1, 2, 3, 4, 6, 8, 12, 24. The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, 36. The greatest common factor is 12.
Now plug into the formula:
LCM(24, 36) = (24 × 36) ÷ 12 = 864 ÷ 12 = 72
Same answer. Different route.
Common Mistakes and What People Get Wrong
I know it sounds simple — but it's easy to mess this up. Here are the traps people fall into:
Confusing LCM with GCD
These are related but opposite concepts. The GCD is the largest number that divides both evenly. Worth adding: the LCM is the smallest number that both divide into evenly. Mixing them up leads to wrong answers, especially when working with fractions.
Forgetting to Take the Highest Power
In the prime factorization method, people sometimes take the lowest power instead of the highest. That gives you 4 × 3 = 12, which is way too small. As an example, they might use 2² instead of 2³, or 3¹ instead of 3². Remember: highest powers only.
Stopping Too Early
Some people list a few multiples and stop before finding the actual LCM. With 24 and 36, you might jump to 144 (24 × 6 or 36 × 4) and think that's the answer. But 72 comes first. Always keep going until you find the smallest match.
Overcomplicating Simple Problems
If the numbers are small enough, just list the multiples. Don't pull out the prime factorization machine for something you can eyeball in 10 seconds. Save the heavy methods for bigger numbers.
Practical Tips That Actually Work
Real talk — here are the strategies that save time and reduce errors:
Use the Right Method for the Numbers
Small numbers (under 50)? Listing multiples is fine. And medium numbers (50–200)? Prime factorization. Large numbers? Definitely use the GCD formula or a calculator.
Double-Check with Division
Once you think you've found the LCM, divide it by both original numbers. Now, both whole numbers. If you get whole numbers both times, you're probably right. For 72: 72 ÷ 24 = 3 and 72 ÷ 36 = 2. Good to go.
Memorize Common Factorizations
Knowing that 24 = 2³ × 3 and 36 = 2² × 3² saves time. Spend a few minutes memorizing the prime factorizations of numbers 1 through 50. It pays off.
Watch for Patterns
If one number is a multiple of the other, the LCM is the larger number. Here's one way to look at it: LCM(12, 24) = 24. This shortcut doesn't apply to 24 and 36, but it's useful to keep in your back pocket.
Use Calculators Wisely
Don't be afraid to use a calculator for the multiplication and division steps. Just make sure you understand the process — don't let the calculator do the thinking for you.
FAQ
What is the LCM of 24 and 36?
The least common multiple of 24 and 36 is 72.
How do you find the LCM step by step?
Use prime factorization: break each number into primes, take the highest power of each prime, then multiply them together.
Why is the LCM important?
It's used for adding fractions, finding common denominators, and determining when repeating events align.
Can you find the LCM without prime factorization?
Yes, by listing multiples
Conclusion
The least common multiple is more than just a math exercise—it’s a practical tool that simplifies complex problems, from aligning schedules to solving fraction equations. By understanding the methods and avoiding common pitfalls, you can approach LCM calculations with confidence. Remember, whether you’re using prime factorization, listing multiples, or leveraging tools like the GCD formula, the key lies in consistency and attention to detail. With practice, these techniques become second nature, turning a seemingly daunting task into an efficient and error-free process. So next time you encounter numbers like 24 and 36, you’ll know exactly how to find their LCM—and avoid the traps that trip up even seasoned learners.
