lcm of 6 24 32

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24-08-2024

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Finding the Least Common Multiple (LCM) of a set of numbers is a fundamental concept in mathematics, especially useful in problems involving fractions, ratios, and synchronizing events. In this article, we will explore how to find the LCM of the numbers 6, 24, and 32.

What is LCM?

The LCM of two or more integers is the smallest positive integer that is divisible by each of the integers. To find the LCM, we can use various methods, including:

• Prime Factorization
• Listing Multiples
• Using the Greatest Common Divisor (GCD)

Step 1: Prime Factorization

Let's start by finding the prime factorization of each number.

1. Prime Factorization of 6

   • (6 = 2^1 \times 3^1)

2. Prime Factorization of 24

   • (24 = 2^3 \times 3^1)

3. Prime Factorization of 32

   • (32 = 2^5)

Step 2: Identify the Highest Powers

To find the LCM using prime factorization, we take the highest power of each prime number that appears in the factorizations:

• For the prime 2: The highest power is (2^5) (from 32).
• For the prime 3: The highest power is (3^1) (from both 6 and 24).

Step 3: Calculate the LCM

Now, we multiply these highest powers together to find the LCM:

[ \text{LCM} = 2^5 \times 3^1 ]

Calculating this gives:

[ \text{LCM} = 32 \times 3 = 96 ]

Conclusion

The Least Common Multiple (LCM) of 6, 24, and 32 is 96. This means that 96 is the smallest number that can be evenly divided by 6, 24, and 32. This concept is useful in many mathematical applications, such as adding fractions with different denominators or finding common periods in cyclic events.