LCM of 6 and 10 by Listing, Division, and Prime Factorization Methods
The concept of LCM of 6 and 10 is essential in mathematics and helps in solving real-world and exam-level problems efficiently. Calculating the Least Common Multiple (LCM) helps with addition and subtraction of fractions, word problems, and time-based questions.
Understanding LCM of 6 and 10
The LCM of 6 and 10 refers to the smallest number that is a multiple of both 6 and 10. This concept is widely used in finding least common multiples, common denominators for fractions, and solving time-interval or scheduling problems. Knowing how to calculate the LCM helps you work faster and with greater accuracy in competitive exams and school tests.
Step-by-Step Methods to Find LCM of 6 and 10
There are three popular ways to find the LCM of 6 and 10:
1. Listing Multiples Method Write out multiples of each number and find the first one they share:Multiples of 10: 10, 20, 30, 40, 50, ...
Common multiples: 30, 60, 90, ...
The LCM of 6 and 10 is 30 (the least).
2. Prime Factorization Method Break both numbers into their prime factors and use each factor the greatest number of times it occurs in either number:
10 = 2 × 5
LCM = 2 × 3 × 5 = 30
3. Division (or Short Division) Method Divide both numbers by common prime factors step by step until both become 1, then multiply all divisors used.
6 ÷ 2 = 3, 10 ÷ 2 = 5
Step 2: Divide by 3:
3 ÷ 3 = 1 (5 not divisible)
Step 3: Divide by 5:
5 ÷ 5 = 1
Multiply divisors: 2 × 3 × 5 = 30
Multiples of 6 and 10 Table
Here’s a helpful table showing the first few multiples to visually confirm the LCM of 6 and 10:
LCM of 6 and 10 Table
| Number | First 5 Multiples |
|---|---|
| 6 | 6, 12, 18, 24, 30 |
| 10 | 10, 20, 30, 40, 50 |
This table shows that 30 is the first common multiple, confirming LCM(6, 10) = 30.
Worked Example – Finding LCM of 6 and 10
Question: Find the LCM of 6 and 10 using the prime factorization method.
6 = 2 × 3
10 = 2 × 5
2. List all factors, using each the greatest number of times it occurs in either factorization:
Prime factors needed: 2 (once), 3 (once), 5 (once)
3. Multiply all these together:
2 × 3 × 5 = 30
Final Answer: LCM(6, 10) = 30
Practice Problems
- Find the LCM of 6, 10, and 15.
- List all multiples of 6 and 10 between 30 and 90.
- Is 60 a common multiple of 6 and 10?
- What is the HCF of 6 and 10? (Hint: try the HCF method, too!)
Common Mistakes to Avoid
- Confusing LCM of 6 and 10 with their highest common factor (HCF).
- Adding or multiplying the numbers instead of finding the least common multiple.
- Missing a factor in the prime factorization method (always use each prime the greatest number of times it appears).
Real-World Applications
The concept of LCM of 6 and 10 appears in real life when matching time intervals (like bus timetables), grouping or packaging items, and solving fraction sums. For example, if two events repeat every 6 and 10 minutes, both will coincide after 30 minutes. Vedantu helps students see how maths applies beyond the classroom for better exam and practical success.
Page Summary
We explored the idea of the LCM of 6 and 10, learned different ways to find it, solved examples, and understood its usefulness in exams and life. Keep practicing with Vedantu for increased confidence and speed in all LCM questions.
Learn More and Practice
Review more about multiples and factors at Multiples, practice LCM with more numbers at LCM of Two Numbers, or see steps for the LCM by Prime Factorization Method. To connect this with fractions, check out Factors of 6 and Factors of 10. Apply these ideas to real problems using Application of LCM and HCF.
FAQs on How to Find the LCM of 6 and 10
1. What is the LCM of 6 and 10?
The LCM (Least Common Multiple) of 6 and 10 is 30. It is the smallest number that is exactly divisible by both 6 and 10, making it essential for solving problems involving multiples and fractions.
2. How to calculate LCM of 6 and 10 using prime factorization?
To find the LCM of 6 and 10 using prime factorization, follow these steps:
1. Factorize each number into its prime factors: 6 = 2 × 3, 10 = 2 × 5.
2. Take the highest powers of each prime factor: 2, 3, and 5.
3. Multiply these primes: 2 × 3 × 5 = 30.
Hence, the LCM is 30.
3. What are the common multiples of 6 and 10?
The common multiples of 6 and 10 are numbers that both 6 and 10 divide exactly. The first few common multiples are: 30, 60, 90, 120, 150, and so on. The smallest common multiple is the LCM, which is 30.
4. What is the HCF of 6 and 10?
The HCF (Highest Common Factor) of 6 and 10 is 2. It is the largest number that divides both 6 and 10 without leaving a remainder, often used alongside LCM to solve various arithmetic problems.
5. How to find LCM of 6, 10, and 15?
To find the LCM of 6, 10, and 15:
1. Find prime factors:
- 6 = 2 × 3
- 10 = 2 × 5
- 15 = 3 × 5
2. Take the highest powers of each prime: 2, 3, and 5.
3. Multiply: 2 × 3 × 5 = 30.
So, the LCM of 6, 10, and 15 is 30.
6. How is LCM used in real life or exams?
The LCM is widely used to solve practical problems such as scheduling events, adding or subtracting fractions with different denominators, and calculating time intervals in exams and daily life scenarios.
7. Why is 12 not the LCM of 6 and 10?
Although 12 is a multiple of 6, it is not divisible by 10. The LCM must be divisible by both numbers. Hence, 12 cannot be the LCM of 6 and 10.
8. Why do students confuse LCM of 6 and 10 with their sum or product?
Students often mistake the LCM for the sum (6 + 10 = 16) or product (6 × 10 = 60) of the numbers. Understanding that the LCM is the smallest common multiple helps avoid this confusion by focusing on divisibility rather than addition or multiplication.
9. Can LCM be smaller than either of the numbers?
No, the LCM cannot be smaller than either of the numbers because it is a multiple of both. The smallest multiple for any number is the number itself, so the LCM is at least as large as the largest number involved.
10. How does LCM change when a third number is added (e.g., 6, 10, and 15)?
When a third number is added, the LCM is recalculated to include the prime factors of all three numbers. It may stay the same or increase, depending on the new number's factors. For example, LCM of 6 and 10 is 30; adding 15 (which factors as 3 × 5) does not change the LCM as it already contains these primes.
11. Is knowing only the factors enough to get the LCM?
Knowing the factors is important but not sufficient alone to find the LCM. You must consider the highest powers of prime factors present in each number to accurately calculate the LCM using prime factorization or division methods.
