Stepwise Calculation of LCM of 12 and 15 Using Prime Factorization
The concept of LCM of 12 and 15 is essential in mathematics and helps in solving real-world and exam-level problems efficiently. Understanding how to find and use the Least Common Multiple can make working with fractions, time schedules, and board exam questions much simpler.
Understanding LCM of 12 and 15
A Least Common Multiple (LCM) of 12 and 15 is the smallest number that is exactly divisible by both 12 and 15 without leaving a remainder. This concept is widely used in fraction addition, time management, and number theory. It plays a vital role in simplifying problems where we need a common base, such as finding a common denominator or arranging things in uniform groups.
Formula Used in LCM of 12 and 15
The standard formula is: \( \text{LCM}(a, b) = \frac{a \times b}{\text{GCF}(a, b)} \), where GCF stands for Greatest Common Factor.
Here’s a helpful table to understand LCM of 12 and 15 more clearly:
LCM of 12 and 15 Table
| Number | Multiples of 12 | Multiples of 15 | Common? |
|---|---|---|---|
| 1 | 12 | 15 | No |
| 2 | 24 | 30 | No |
| 3 | 36 | 45 | No |
| 4 | 48 | 60 | Yes (LCM) |
| 5 | 60 | 75 | Yes (LCM) |
This table shows how the pattern of the LCM of 12 and 15 appears regularly in real cases. The smallest common multiple is 60.
Step-by-Step: How to Find LCM of 12 and 15
Let’s calculate LCM of 12 and 15 using two popular methods: Prime Factorization and Division (Long Division).
Method 1: Prime Factorization
1. Write the prime factors for each number.
15 = 3 × 5 = \( 3^1 \times 5^1 \)
2. Take the highest power of each prime number found in either number:
Highest power of 3 = \( 3^1 \)
Highest power of 5 = \( 5^1 \)
3. Multiply these together:
Method 2: Long Division Method
1. Write 12 and 15 side by side.
2. Divide by 2:
3. Divide by 2 again:
4. Divide by 3:
5. Divide by 5:
6. Multiply all the divisors used:
Worked Example – Finding LCM for Other Numbers
Question: What is the LCM of 12, 15, and 10?
Step 1: Find LCM of 12 and 15 (as shown above): 60
Step 2: Find LCM of 60 and 10.
Prime factors: 60 = 2 × 2 × 3 × 5, 10 = 2 × 5
Take the highest power of each:
22, 3, 5
LCM = 2 × 2 × 3 × 5 = 60
So, the LCM of 12, 15, and 10 is also 60.
LCM and HCF: What’s the Difference?
| Term | Full Form | Description | Example (12 & 15) |
|---|---|---|---|
| LCM | Least Common Multiple | Smallest number exactly divisible by both numbers | 60 |
| HCF | Highest Common Factor | Greatest number that divides both numbers | 3 |
Remember, HCF is about division (making groups), and LCM is about making things match up in multiples.
Real-World Applications
The concept of LCM of 12 and 15 appears in areas such as adding or subtracting unlike fractions, creating schedules or timetables, and in solving maths puzzles. Applications of LCM and HCF are explained in detail for practical problem solving. Vedantu helps students see how maths applies beyond the classroom.
Practice Problems
- Find the LCM of 12 and 15 by listing all their multiples.
- Is 30 the LCM of 12 and 15?
- What is the HCF of 12 and 15? Explain your answer.
- List all common multiples of 12 and 15 up to 120.
Common Mistakes to Avoid
- Confusing LCM with HCF – remember, the LCM is always bigger unless the numbers are the same.
- Missing out prime factors when using the prime factorization method.
- Thinking the product of the numbers is always the LCM (it’s not—only if they are co-prime).
Revision Table: Quick Facts
| Step | Action | Result |
|---|---|---|
| Prime Factors (12) | 2 × 2 × 3 | 2² × 3 |
| Prime Factors (15) | 3 × 5 | 3 × 5 |
| Highest Powers | 2², 3, 5 | All taken |
| LCM | Multiply them | 60 |
Explore Further
- LCM – In-depth concept and more examples
- Factors of 12 – Learn all divisors of 12
- Factors of 15 – Necessary for LCM and HCF
- LCM by Prime Factorization Method – Stepwise guides
- LCM by Long Division Method – Visual steps for division method
- HCF by Long Division Method – See how HCF and LCM relate
- Application of LCM and HCF – Real-world and word problems
- Multiples of 4 – How multiples work in LCM
- Common Multiple – Meaning and examples
- Multiples – Foundation for LCM
- Factors of a Number – Learn factorization basics
We explored the idea of LCM of 12 and 15, how to apply it, solve related problems, and understand its real-life relevance. Practice more with Vedantu to build confidence in these concepts and excel in your exams.
FAQs on How to Find the LCM of 12 and 15
1. What is the LCM of 12 and 15?
The LCM (Least Common Multiple) of 12 and 15 is 60. It is the smallest positive integer that is divisible by both 12 and 15 without any remainder.
2. How to find the LCM of 12 and 15 by prime factorization?
To find the LCM of 12 and 15 by prime factorization, follow these steps:
1. Find the prime factors of each number: 12 = 22 × 3, 15 = 3 × 5.
2. Take the highest power of each prime factor appearing in both: 22, 3, and 5.
3. Multiply these together: 22 × 3 × 5 = 4 × 3 × 5 = 60.
Thus, the LCM is 60.
3. What is the LCM of 12, 15 and 10?
The LCM of 12, 15 and 10 is found by first determining the prime factors: 12 = 22 × 3, 15 = 3 × 5, 10 = 2 × 5.
Taking the highest powers: 22, 3, and 5.
Multiply them: 4 × 3 × 5 = 60.
Therefore, the LCM of 12, 15 and 10 is also 60.
4. How is LCM different from HCF/GCF?
The LCM (Least Common Multiple) is the smallest multiple common to both numbers, while the HCF (Highest Common Factor), also called GCF (Greatest Common Factor), is the largest number that divides both numbers exactly.
For example, for 12 and 15:
- LCM = 60
- HCF/GCF = 3.
This difference is important in solving fraction problems and simplifies calculations involving multiples and factors.
5. Why is finding the LCM important in maths?
Finding the LCM is crucial because it helps in:
- Adding or subtracting fractions with different denominators by finding a common denominator.
- Solving problems related to time, such as scheduling and repeating events.
- Simplifying problems in number theory, algebra, and competitive exams.
It provides a foundation for many mathematical concepts and real-life applications.
6. Why isn’t the product of 12 and 15 always the LCM?
The product of two numbers is not always their LCM because the numbers may share common factors. The LCM is the product of the numbers divided by their HCF. For 12 and 15, product = 180, but their HCF is 3, so:
LCM = (12 × 15) / 3 = 180 / 3 = 60.
This adjustment prevents double-counting common factors.
7. Why do students mix up LCM and GCF in exam problems?
Students often confuse LCM and GCF (Greatest Common Factor) because both involve finding common values between numbers. However:
- LCM is about multiples (the smallest number divisible by both).
- GCF is about factors (the largest number dividing both).
Clear understanding and practice differentiating multiples vs factors help avoid this confusion.
8. If two numbers are co-prime, what is their LCM?
Two numbers are co-prime if their GCF (HCF) is 1, meaning they have no common prime factors.
For co-prime numbers, the LCM is simply their product.
Example: 12 and 35 are co-prime, so LCM = 12 × 35 = 420.
9. Are there tricks to finding LCM fast for three numbers?
Yes, some fast methods include:
1. Finding the LCM of two numbers first, then finding the LCM of that result with the third number.
2. Using prime factorization for all three and taking the highest powers.
3. Using the formula: LCM(a,b,c) = LCM(LCM(a,b), c).
These tricks help save time during exams.
10. Can LCM be used for decimals or only whole numbers?
LCM is typically defined for whole numbers only. But to find LCM of decimals:
- Convert decimals to whole numbers by multiplying with powers of 10.
- Find the LCM of these whole numbers.
- Divide back by the same power of 10.
This method helps in applying LCM concepts to decimals in practical problems.
