LCM Calculator
Least Common Multiple
[Image of least common multiple Venn diagram]
$$ \text{lcm}(A, B) = ? $$
Number A
Number B
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Least Common Multiple
Calculation Steps
👨🏫
By Prof. David Anderson
Mathematics Professor | 20+ Years Exp.
“Have you ever tried to add $\frac{1}{4} + \frac{1}{6}$ and got stuck? You’re not alone. The secret weapon you are missing is the LCD (Least Common Denominator), which is really just the LCM in disguise. In my 20 years of teaching, I’ve found that the best way to understand this is to see the numbers jumping along the number line until they meet. Let’s make them jump!”
LCM Calculator (Least Common Multiple)
Find LCD for Fractions & Listing Multiples Step-by-Step
Method 1: Visualizing the “Meeting Point”
Scroll right to find the first matching green number!
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1. What is the Least Common Multiple?
Imagine two runners on a circular track. Runner A completes a lap every 4 minutes. Runner B takes 6 minutes. When will they cross the starting line together again?
- Runner A hits the line at: 4, 8, 12, 16, 20, 24...
- Runner B hits the line at: 6, 12, 18, 24...
They sync up at 12 minutes, 24 minutes, etc. The first time they meet (12) is the Least Common Multiple (LCM). In math, "Least" means smallest, "Common" means shared, and "Multiple" means the result of multiplying.
2. The LCD Connection: Fractions Made Easy
Why do we learn this? For Fractions!
You cannot add apples to oranges. Similarly, you cannot add fractions with different denominators (bottom numbers). You need a Common Denominator.
Problem: $$ \frac{1}{4} + \frac{1}{6} $$
The denominators are 4 and 6. The Least Common Denominator (LCD) is simply $LCM(4, 6) = 12$.
Convert:
• $\frac{1 \times 3}{4 \times 3} = \frac{3}{12}$
• $\frac{1 \times 2}{6 \times 2} = \frac{2}{12}$
Result: $\frac{3}{12} + \frac{2}{12} = \frac{5}{12}$.
3. Three Ways to Calculate LCM
Method A: The Listing Method (The Visual Way)
Best for small numbers (like 4 and 6). This is exactly what our tool above does.
1. List the multiples of the first number.
2. List the multiples of the second number.
3. Circle the first number that appears in both lists.
Method B: Prime Factorization (The Venn Way)
Best for understanding the "DNA" of the numbers.
Example: LCM(12, 18)
• $12 = 2^2 \times 3$
• $18 = 2 \times 3^2$
To find LCM, take the highest power of every prime factor present.
• Highest 2: $2^2$ (from 12)
• Highest 3: $3^2$ (from 18)
• $LCM = 2^2 \times 3^2 = 4 \times 9 = 36$.
Method C: The Formula (Using GCD)
Best for computer science and huge numbers. If you already know the GCD (Greatest Common Divisor), use this shortcut:
$$ LCM(a, b) = \frac{|a \times b|}{GCD(a, b)} $$
Example: Find LCM(12, 18).
Product: $12 \times 18 = 216$.
GCD(12, 18) is 6.
LCM = $216 \div 6 = 36$.
4. Real World Scenarios
🌭 Scenario 1: The Hot Dog Problem
Hot dogs come in packs of 10. Buns come in packs of 8. What is the minimum number of packs you must buy to have exactly one bun for every hot dog with none left over?
Solution: Find $LCM(10, 8)$.
Multiples of 10: 10, 20, 30, 40...
Multiples of 8: 8, 16, 24, 32, 40...
LCM is 40. You need 40 hot dogs (4 packs) and 40 buns (5 packs).
🪐 Scenario 2: Planetary Alignment
Planet A orbits the sun every 10 years. Planet B orbits every 15 years. If they align today, when will they align again?
Solution: Find $LCM(10, 15)$.
10: 10, 20, 30...
15: 15, 30...
Answer: In 30 years.
Comparison: LCM vs GCD
| Feature | GCD (Greatest Common Divisor) | LCM (Least Common Multiple) |
|---|---|---|
| Direction | Breaking Down (Division) | Building Up (Multiplication) |
| Key Use | Simplifying Fractions | Adding Fractions (LCD) |
| Size | Smaller than or equal to numbers | Larger than or equal to numbers |
| Example (4, 6) | 2 | 12 |
5. Professor's FAQ Corner
Can LCM be smaller than the numbers?
Never. Since it is a "multiple", it must be at least as large as the largest number in the set. For example, LCM(5, 10) is 10.
How do I find the LCM of 3 numbers?
Just like with 2 numbers! Find the LCM of the first two, then find the LCM of that result and the third number.
$LCM(a, b, c) = LCM(LCM(a, b), c)$.
$LCM(a, b, c) = LCM(LCM(a, b), c)$.
What is the LCM of prime numbers?
If the numbers are prime (or co-prime), the LCM is simply their product. $LCM(3, 5) = 3 \times 5 = 15$.
Is LCM the same as LCD?
Yes and no. The math is identical. "LCD" is just the specific name we use when we apply the LCM process to the denominators of fractions.
References
- Euclid. Elements (Book VII). c. 300 BC. (Foundational number theory).
- Hardy, G. H., & Wright, E. M. (2008). An Introduction to the Theory of Numbers. Oxford University Press.
- Knuth, D. E. (1997). The Art of Computer Programming. (Algorithms for GCD and LCM).
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