LCM of 8 and 12 by Prime Factorization Method
The concept of LCM of 8 and 12 is a vital part of arithmetic, especially for students preparing for school exams and competitive tests like JEE and NEET. Understanding how to find the Least Common Multiple (LCM) helps you solve problems in fractions, algebra, scheduling, and many real-life scenarios. Mastering this topic ensures you can handle number properties and applications confidently.
Understanding LCM of 8 and 12
The LCM (Least Common Multiple) of two numbers is the smallest number that both numbers divide exactly (without leaving any remainder). It is especially useful for operations involving addition, subtraction, or comparison of fractions as well as for solving word problems involving repeating events. For the numbers 8 and 12, the LCM is the smallest number that both 8 and 12 multiply into evenly. Grasping this core concept helps with many arithmetic and algebraic operations.
Multiples of 8 and 12: Visualizing the Intersection
To better understand the idea of common multiples, let's list the first few multiples of each number:
| Multiples of 8 | Multiples of 12 |
|---|---|
| 8, 16, 24, 32, 40, 48, 56, 64 | 12, 24, 36, 48, 60, 72, 84, 96 |
As you can see, the smallest number that appears in both lists is 24. This is the LCM of 8 and 12.
Methods to Find LCM of 8 and 12
There are several ways to find the LCM, enabling all students to pick the method that makes the most sense to them. Let’s explore three common methods:
-
Listing Multiples Method
List the multiples of both numbers and find the first number common to both lists.
Multiples of 8: 8, 16, 24, 32, 40, 48...
Multiples of 12: 12, 24, 36, 48...
The smallest common multiple is 24. -
Prime Factorization Method
- Break down both numbers into their prime factors:
- 8 = 2 × 2 × 2 = 23
- 12 = 2 × 2 × 3 = 22 × 3
- Select the highest powers of all the prime numbers found:
- LCM = 23 × 31 = 8 × 3 = 24
-
Division (Short Division) Method
Arrange the numbers side by side and divide by the smallest prime number that can divide at least one of them. Continue dividing until only 1’s remain:
Step 8 12 Prime Divisor 1 8 12 2 2 4 6 2 3 2 3 2 4 1 3 3 5 1 1 Multiply all the prime divisors: 2 × 2 × 2 × 3 = 24
Worked Examples
Let’s solve for the LCM of 8 and 12 using each method step-by-step:
-
By Listing Multiples:
Multiples of 8: 8, 16, 24, 32, 40, ...
Multiples of 12: 12, 24, 36, ...
LCM = 24 -
By Prime Factorization:
8 = 2 × 2 × 2 = 23
12 = 2 × 2 × 3 = 22 × 3
LCM = (use the highest powers) = 23 × 31 = 8 × 3 = 24 -
By Division Method:
Divide by 2: (8, 12) → (4, 6)
Divide by 2: (4, 6) → (2, 3)
Divide by 2: (2, 3) → (1, 3)
Divide by 3: (1, 3) → (1, 1)
Multiply prime divisors: 2 × 2 × 2 × 3 = 24
Practice Problems
- Find the LCM of 10 and 12.
- Find the LCM of 8, 12, and 16.
- Which is larger, the LCM of 6 and 8, or the LCM of 8 and 12?
- Two bells ring at intervals of 8 and 12 minutes. After how many minutes will they ring together?
- If you want to add the fractions 5/8 and 7/12, what denominator should you use?
Common Mistakes to Avoid
- Confusing LCM with HCF/GCD. Remember, LCM is about multiples, not greatest common divisors.
- Missing out on using the HIGHEST powers of prime factors for LCM.
- Not checking the smallest common multiple correctly when listing multiples.
- Stopping the short division method too early, resulting in an incorrect answer.
Real-World Applications
The LCM of 8 and 12 is handy in real life. For example, if two buses arrive every 8 and 12 minutes, the next time they both arrive together is after 24 minutes. LCM is also used for solving problems with repeat cycles, finding common denominators in fractions, and scheduling tasks or events. At Vedantu, we connect maths problems like these to practical solutions for students’ daily uses.
Page Summary
In this topic, we learned to find the LCM of 8 and 12 using three methods: listing multiples, prime factorization, and the division method. We practiced stepwise solutions and tackled application problems. Understanding LCM is key to solving many number theory and real-life problems, and with the right methods, every student can master it. For more about LCM, multiples, factors, and practice worksheets, keep exploring Vedantu’s maths section.
Looking to explore further? Check out related topics:
What is LCM? Detailed Concept & Formulas |
Factors of 8 |
Factors of 12 |
LCM by Division Method |
Prime Factorization: Steps & Examples |
HCF of 8 and 12 |
Multiples of 8 |
Multiples of 12
FAQs on How to Find the LCM of 8 and 12
1. What is the LCM of 8 and 12?
The Lowest Common Multiple (LCM) of 8 and 12 is 24. This is the smallest number that is a multiple of both 8 and 12.
2. How do you find the LCM by prime factorization?
To find the LCM using prime factorization:
• First, find the prime factors of each number: 8 = 2 x 2 x 2 and 12 = 2 x 2 x 3.
• Identify the highest power of each prime factor present in either number: 23 and 31.
• Multiply these highest powers together: 2 x 2 x 2 x 3 = 24. Therefore, the LCM of 8 and 12 is 24.
3. What is the difference between LCM and HCF?
The LCM (Lowest Common Multiple) is the smallest number that is a multiple of both numbers, while the HCF (Highest Common Factor) is the largest number that divides both numbers without leaving a remainder. For 8 and 12, the LCM is 24 and the HCF is 4.
4. Why do we calculate LCM?
We use LCM to solve problems involving fractions, ratios, and finding common cycles or patterns. For instance, determining when two events will occur simultaneously. It's frequently applied in word problems concerning time, schedules, and measurements.
5. Can the LCM of two numbers ever be less than both?
No, the LCM of two numbers can never be less than either of the two numbers. The LCM is always greater than or equal to the larger of the two numbers.
6. What is the LCM and HCF of 8 and 12?
The LCM of 8 and 12 is 24, and the HCF (Highest Common Factor, also known as GCF - Greatest Common Factor) is 4.
7. How do you calculate LCM?
You can calculate the LCM (Least Common Multiple) using several methods: listing multiples, prime factorization, or the division method. The method you choose depends on the complexity of the numbers and your preference. The result, however, is always the same.
8. Why is the LCM of 8 and 12 24?
The LCM of 8 and 12 is 24 because 24 is the smallest positive integer that is divisible by both 8 and 12 without any remainder. It's the least common multiple among all their multiples.
9. What is the LCM calculator?
An LCM calculator is a tool, either online or on a physical device, that automatically calculates the Least Common Multiple of two or more numbers. It saves time and effort in calculating the LCM, especially for larger numbers.
10. LCM of 8 and 12 by division method?
Using the division method:
• Divide both numbers by their common prime factors until you reach 1.
• Multiply all the divisors to find the LCM. For 8 and 12, this process yields 2 x 2 x 2 x 3 = 24. Therefore, the LCM of 8 and 12 is 24.
11. LCM of 8 and 12 using prime factorization?
Using prime factorization:
• Find the prime factors of each number: 8 = 2 x 2 x 2, 12 = 2 x 2 x 3
• Take the highest power of each prime factor: 23 and 31.
• Multiply these together: 2 x 2 x 2 x 3 = 24. This is the LCM of 8 and 12.
12. What are multiples of 8 and 12?
Multiples of 8: 8, 16, 24, 32, 40, 48, 56, 64...
Multiples of 12: 12, 24, 36, 48, 60, 72, 84, 96... The LCM is the smallest number common to both lists (24).
