In this article, we will break down how to calculate the LCM of 18, 24, and 42 using simple and effective methods. The Least Common Multiple (LCM) is the smallest number that is a multiple of two or more numbers. Let’s dive into the steps to find the LCM of these numbers!
Steps to Find the LCM of 18, 24, and 42
We’ll use the prime factorization method to calculate the LCM, which is one of the most efficient ways to find it.
Step 1: Prime Factorization of Each Number
First, we need to find the prime factors of each number.
- Prime factorization of 18:
18 = 2×3×3 = 2×32 - Prime factorization of 24:
24 = 2 × 2 × 2 × 3 = 23 × 3 - Prime factorization of 42:
42 = 2 × 3 × 7 = 2 × 3 × 7
Step 2: Identify the Highest Powers of Prime Factors
To find the LCM, we take the highest power of each prime number that appears in the prime factorizations.
- The prime factors are 2, 3, and 7.
- The highest power of 2 is ( 23 ) (from 24).
- The highest power of 3 is ( 32 ) (from 18).
- The highest power of 7 is ( 7 ) (from 42).
Step 3: Multiply the Highest Powers Together
Now, we multiply these together to find the LCM:
LCM = 23 × 32 × 7 = 8 × 9 × 7
LCM = 72 × 7 = 504
So, the LCM of 18, 24, and 42 is 504.
Verification of the LCM
To confirm, let’s list the multiples of 18, 24, and 42 and check if 504 is indeed the smallest common multiple.
- Multiples of 18: 18, 36, 54, 72, …, 504
- Multiples of 24: 24, 48, 72, 96, …, 504
- Multiples of 42: 42, 84, 126, 168, …, 504
Since 504 is the first common multiple in all three lists, the LCM is correct.
Conclusion
The LCM of 18, 24, and 42 is 504. This method of using prime factorization ensures accuracy and efficiency when finding the Least Common Multiple of multiple numbers. Whether you’re solving math problems or handling real-life applications like scheduling or managing frequencies, knowing how to calculate the LCM can be incredibly useful.
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