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How To Find Lcm Of 8 And 12 One method is to list out the multiples of each number and find the first common multiple which is LCM. Another method is to use prime factorization to find the highest power of each prime factor that appears in either number and multiply those together to get the LCM. These methods are also used in How To Find Lcm Of 8 And 12. If you are searching for How To Find Lcm Of 8 And 12, Read the content below.
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Contents
How To Find Lcm Of 8 And 12?
The LCM or Least Common Multiple of two or more numbers is the smallest number that is a multiple of all the given numbers. In this case, we need to find the LCM of 8 and 12.
To find the LCM, we first list the multiples of each number, which are the numbers that can be obtained by multiplying the number by any integer. For example, the multiples of 8 are 8, 16, 24, 32, 40, 48, and so on. Similarly, the multiples of 12 are 12, 24, 36, 48, 60, and so on.
Next, we identify the common multiples of both numbers, which are the numbers that appear in both lists. In this case, the common multiples of 8 and 12 are 24, 48, 72, and 96.
Finally, we choose the smallest common multiple as the LCM. So, in this case, the smallest common multiple is 24, which means that 24 is the smallest number that is a multiple of both 8 and 12. Therefore, the LCM of 8 and 12 is 24.
To find the LCM (Least Common Multiple) of 8 and 12, you can follow these steps:
Step 1: List the multiples of each number
Multiples of 8: 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96, …
Multiples of 12: 12, 24, 36, 48, 60, 72, 84, 96, …
Step 2: Identify the common multiples of both numbers
The common multiples of 8 and 12 are: 24, 48, 72, 96, …
Step 3: Identify the smallest common multiple
The smallest common multiple is 24, so the LCM of 8 and 12 is 24.
Therefore, the LCM of 8 and 12 is 24.
Example of Lcm Of 8 And 12
Here are some examples of finding the LCM of two or more numbers:
Example 1: Find the LCM of 3 and 5
Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, …
Multiples of 5: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, …
The common multiples of 3 and 5 are 15, 30, 45, 60, …
The smallest common multiple is 15, so the LCM of 3 and 5 is 15.
Example 2: Find the LCM of 4, 6, and 8
Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, …
Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, …
Multiples of 8: 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, …
The common multiples of 4, 6, and 8 are 24, 48, and 72.
The smallest common multiple is 24, so the LCM of 4, 6, and 8 is 24.
Example 3: Find the LCM of 10, 15, and 20
Multiples of 10: 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, …
Multiples of 15: 15, 30, 45, 60, 75, 90, 105, 120, …
Multiples of 20: 20, 40, 60, 80, 100, 120, 140, …
The common multiples of 10, 15, and 20 are 60 and 120.
The smallest common multiple is 60, so the LCM of 10, 15, and 20 is 60.
What Is The Lcm Of 8 And 12?
To find the LCM (Least Common Multiple) of 8 and 12, we need to list the multiples of both numbers and identify the smallest number that appears in both lists. Here are the multiples of 8 and 12:
Multiples of 8: 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96, …
Multiples of 12: 12, 24, 36, 48, 60, 72, 84, 96, …
We can see that the smallest number that appears in both lists is 24. Therefore, the LCM of 8 and 12 is 24.
So, the LCM of 8 and 12 is 24.
Here are a few more examples of finding the LCM of two numbers:
Example 1: Find the LCM of 6 and 9
Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, …
Multiples of 9: 9, 18, 27, 36, 45, 54, 63, 72, 81, 90, …
The smallest number that appears in both lists is 18, so the LCM of 6 and 9 is 18.
Example 2: Find the LCM of 12 and 20
Multiples of 12: 12, 24, 36, 48, 60, 72, 84, 96, …
Multiples of 20: 20, 40, 60, 80, 100, 120, …
The smallest number that appears in both lists is 60, so the LCM of 12 and 20 is 60.
Example 3: Find the LCM of 7 and 15
Multiples of 7: 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, …
Multiples of 15: 15, 30, 45, 60, 75, 90, 105, 120, …
The smallest number that appears in both lists is 105, so the LCM of 7 and 15 is 105.
Here are the steps to find the LCM (Least Common Multiple) of 8 and 12:
Step 1: Write out the multiples of each number until you find a multiple that is common to both.
Multiples of 8: 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, …
Multiples of 12: 12, 24, 36, 48, 60, 72, 84, 96, …
Step 2: Identify the smallest number that appears in both lists. This is the LCM.
In this case, the smallest number that appears in both lists is 24, so the LCM of 8 and 12 is 24.
That’s it! So, the LCM of 8 and 12 is 24.
Examples of Lcm Of 8 And 12
Here are some illustrations of finding the LCM of two numbers:
Illustration 1: Find the LCM of 6 and 8
Step 1: Write out the multiples of each number until you find a multiple that is common to both.
Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48, …
Multiples of 8: 8, 16, 24, 32, 40, 48, …
Step 2: Identify the smallest number that appears in both lists. This is the LCM.
In this case, the smallest number that appears in both lists is 24, so the LCM of 6 and 8 is 24.
Illustration 2: Find the LCM of 10 and 15
Step 1: Write out the multiples of each number until you find a multiple that is common to both.
Multiples of 10: 10, 20, 30, 40, 50, 60, …
Multiples of 15: 15, 30, 45, 60, 75, …
Step 2: Identify the smallest number that appears in both lists. This is the LCM.
In this case, the smallest number that appears in both lists is 30, so the LCM of 10 and 15 is 30.
Illustration 3: Find the LCM of 9 and 12
Step 1: Write out the multiples of each number until you find a multiple that is common to both.
Multiples of 9: 9, 18, 27, 36, 45, …
Multiples of 12: 12, 24, 36, 48, …
Step 2: Identify the smallest number that appears in both lists. This is the LCM.
In this case, the smallest number that appears in both lists is 36, so the LCM of 9 and 12 is 36.
I hope these illustrations help you understand how to find the LCM of two numbers.
What Is The Lcm Of 8 And 12 Using Prime Factorization?
We can find the LCM of 8 and 12 using prime factorization as follows:
Step 1: Find the prime factorization of each number.
The prime factorization of 8 is 2 x 2 x 2.
The prime factorization of 12 is 2 x 2 x 3.
Step 2: Identify the highest power of each prime factor that appears in either factorization, and multiply these factors together.
The highest power of 2 that appears in either factorization is 2 x 2 = 4.
The highest power of 3 that appears in either factorization is 3.
Therefore, the LCM of 8 and 12 is 4 x 3 = 12.
So, the LCM of 8 and 12 using prime factorization is 12.
Prime factorization is a process in which a composite number is expressed as a product of its prime factors. To find the LCM using prime factorization, you first write each number as a product of its prime factors. Then, you identify the prime factors that appear in either number, and combine them by taking the highest power of each prime factor. Finally, you multiply these prime factors together to get the LCM.
For example, let’s say we want to find the LCM of 12 and 20 using prime factorization:
Step 1: Find the prime factorization of each number.
The prime factorization of 12 is 2 x 2 x 3.
The prime factorization of 20 is 2 x 2 x 5.
Step 2: Identify the prime factors that appear in either factorization, and take the highest power of each.
- The prime factors that appear in either factorization are 2, 3, and 5.
- The highest power of 2 is 2 x 2 = 4.
- The highest power of 3 is 3.
- The highest power of 5 is 5.
Step 3: Multiply the prime factors together to get the LCM.
Therefore, the LCM of 12 and 20 using prime factorization is 60.
Examples of Lcm Of 8 And 12 Using Prime Factorization
Here are some examples of finding the LCM using prime factorization:
Example 1: Find the LCM of 6 and 8 using prime factorization.
Step 1: Find the prime factorization of each number.
The prime factorization of 6 is 2 x 3.
The prime factorization of 8 is 2 x 2 x 2.
Step 2: Identify the prime factors that appear in either factorization, and take the highest power of each.
- The prime factors that appear in either factorization are 2 and 3.
- The highest power of 2 is 2 x 2 x 2 = 8.
- The highest power of 3 is 3.
Step 3: Multiply the prime factors together to get the LCM.
Therefore, the LCM of 6 and 8 using prime factorization is 24.
Example 2: Find the LCM of 12 and 18 using prime factorization.
Step 1: Find the prime factorization of each number.
The prime factorization of 12 is 2 x 2 x 3.
The prime factorization of 18 is 2 x 3 x 3.
Step 2: Identify the prime factors that appear in either factorization, and take the highest power of each.
- The prime factors that appear in either factorization are 2 and 3.
- The highest power of 2 is 2 x 2 = 4.
- The highest power of 3 is 3 x 3 = 9.
Step 3: Multiply the prime factors together to get the LCM.
Therefore, the LCM of 12 and 18 using prime factorization is 36.
I hope these examples help you understand how to find the LCM using prime factorization.
Why Is 24 The Lcm Of 8 And 12?
24 is the LCM of 8 and 12 because it is the smallest number that is divisible by both 8 and 12 without leaving any remainder.
To see why 24 is the LCM, we can list the multiples of 8 and 12 and find the first number that they have in common:
Multiples of 8: 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96, …
Multiples of 12: 12, 24, 36, 48, 60, 72, 84, 96, …
From the list, we can see that 24 is the first number that appears in both lists, and therefore it is the smallest common multiple of 8 and 12.
Another way to find the LCM is by using the prime factorization method, which we discussed earlier. In this case, the prime factorization of 8 is 2 x 2 x 2, and the prime factorization of 12 is 2 x 2 x 3. To find the LCM, we take the highest power of each prime factor that appears in either factorization and multiply them together, giving us:
LCM = 2 x 2 x 2 x 3 = 24.
Therefore, 24 is the LCM of 8 and 12.
The LCM (Least Common Multiple) is the smallest positive number that is divisible by two or more given numbers without leaving a remainder. In this case, we want to find the LCM of 8 and 12. To do this, we can start by listing the multiples of each number until we find a common multiple:
Multiples of 8: 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96, …
Multiples of 12: 12, 24, 36, 48, 60, 72, 84, 96, …
We can see that 24 is the smallest number that is in both lists. Therefore, 24 is the LCM of 8 and 12.
Another way to find the LCM is to use prime factorization. We can write each number as a product of its prime factors, which are the building blocks of all numbers:
8 = 2 x 2 x 2
12 = 2 x 2 x 3
We can then find the product of all the prime factors that appear in either number, taking the highest power of each prime factor:
2 x 2 x 2 x 3 = 24
Again, we see that the LCM of 8 and 12 is 24.
Examples of Lcm Of 8 And 12
Here are some new examples of finding the LCM:
Example 1: Find the LCM of 4 and 6.
Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, …
Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48, …
We can see that the first common multiple is 12. Therefore, the LCM of 4 and 6 is 12.
Example 2: Find the LCM of 15 and 20.
Multiples of 15: 15, 30, 45, 60, 75, 90, 105, 120, …
Multiples of 20: 20, 40, 60, 80, 100, 120, …
We can see that the first common multiple is 60. Therefore, the LCM of 15 and 20 is 60.
Example 3: Find the LCM of 24 and 36 using prime factorization.
Prime factorization of 24: 2 x 2 x 2 x 3
Prime factorization of 36: 2 x 2 x 3 x 3
We take the highest power of each prime factor that appears in either factorization and multiply them together:
2 x 2 x 2 x 3 x 3 = 72
Therefore, the LCM of 24 and 36 is 72.
How Do You Find The LCM?
To find the LCM (Least Common Multiple) of two or more numbers, there are several methods you can use. Here are three common methods:
Listing Multiples:
- Start by listing the multiples of each number until you find a common multiple. The first common multiple that you come across is the LCM. This method is simple but can be time-consuming for larger numbers.
Prime Factorization:
- Write each number as a product of its prime factors. To find the LCM, take the highest power of each prime factor that appears in either factorization and multiply them together. This method is efficient and is preferred for larger numbers.
Division Method:
- This method involves using long division to find the LCM. Divide the larger number by the smaller number and write down the quotient and the remainder. Then, divide the smaller number by the remainder and write down the quotient and the remainder. Continue this process until the remainder is zero. The LCM is the product of all the divisors and the last non-zero remainder. This method can be time-consuming but is useful for finding the LCM of more than two numbers.
Each of these methods can be used to find the LCM. The choice of method depends on the size of the numbers and personal preference.
Euclidean Algorithm:
- This method involves using the GCD (Greatest Common Divisor) of the numbers to find the LCM. Divide the product of the numbers by their GCD to get the LCM. This method is fast and efficient, especially for large numbers.
Venn Diagram Method:
- Draw a Venn diagram with circles for each number. List the factors of each number in their respective circles. To find the LCM, multiply all the factors in the diagram. This method is visual and easy to understand but can be impractical for larger numbers.
Matrix Method:
- Write the numbers in a matrix with each row representing a number and each column representing a prime factor. Fill in the matrix with the number of times each prime factor appears in each number’s prime factorization. To find the LCM, multiply the prime factors raised to their highest power in each column. This method is useful for finding the LCM of more than two numbers.
These are just a few additional methods for finding the LCM. The choice of method depends on the numbers and personal preference.
Here are some examples of how to find the LCM of different sets of numbers using various methods:
Listing Multiples:
- To find the LCM of 4 and 6:
- Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52…
- Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48, 54…
- The first common multiple is 12, so LCM(4,6) = 12.
Prime Factorization:
- To find the LCM of 12 and 20:
- Prime factorization of 12: 2^2 * 3^1
- Prime factorization of 20: 2^2 * 5^1
- Take the highest power of each prime factor: 2^2 * 3^1 * 5^1 = 60
- LCM(12,20) = 60.
Division Method:
- To find the LCM of 24 and 36:
- 36 ÷ 24 = 1 R 12
- 24 ÷ 12 = 2 R 0
- The divisors are 2, 3, and 2 (in that order), and the last non-zero remainder is 12.
- LCM(24,36) = 2 × 3 × 2 × 12 = 144.
Euclidean Algorithm:
- To find the LCM of 15 and 24:
- GCD(15,24) = 3
- LCM(15,24) = (15 × 24) ÷ 3 = 120.
Venn Diagram Method:
- To find the LCM of 18, 20, and 24:
- Factors of 18: 2, 3, 6, 9, 18
- Factors of 20: 2, 4, 5, 10, 20
- Factors of 24: 2, 3, 4, 6, 8, 12, 24
- The LCM is the product of all the factors in the Venn diagram, which is 2^2 × 3^1 × 4^1 × 5^1 × 6^1 × 8^1 × 9^1 × 10^1 × 12^1 × 18^1 × 20^1 × 24^1 = 2,160,000.
Matrix Method:
- To find the LCM of 12, 15, and 24:
Matrix:
| | 2 | 3 |
|—|—|—|
| 12| 2 | 1 |
| 15| 1 | 1 |
- | 24| 3 | 1 |
- The LCM is the product of the prime factors raised to their highest power, which is 2^3 × 3^1 × 5^1 = 120.
These are just a few examples of how to find the LCM using different methods.
How To Find Lcm Of 8 And 12 – FAQ
1. What does LCM stand for?
LCM stands for “least common multiple.”
2. Why do we need to find the LCM of two numbers?
We need to find the LCM of two numbers in order to determine the smallest multiple that both numbers have in common.
3. Can we use the listing multiples method to find the LCM of 8 and 12?
Yes, we can use the listing multiples method to find the LCM of 8 and 12.
4. How does the prime factorization method work in finding the LCM of 8 and 12?
The prime factorization method works by finding the prime factors of each number, and then multiplying the highest power of each prime factor that appears in either number to get the LCM.
5. Can the division method be used to find the LCM of 8 and 12?
Yes, the division method can be used to find the LCM of 8 and 12.
6. What is the Euclidean algorithm, and can it be used to find the LCM of 8 and 12?
The Euclidean algorithm is a method of finding the greatest common divisor of two numbers, which can then be used to find the LCM. Yes, the Euclidean algorithm can be used to find the LCM of 8 and 12.
7. What is the Venn diagram method, and how does it work in finding the LCM of 8 and 12?
The Venn diagram method involves drawing a Venn diagram with two circles, one for each number, and filling in the common factors of the two numbers in the overlapping region. The LCM is then found by multiplying the factors in the overlapping region and any remaining factors in the circles.
8. Is the LCM of 8 and 12 unique?
Yes, the LCM of 8 and 12 is unique, as there is only one smallest multiple that both numbers have in common.
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