what is the least common multiple of 6 and 3

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31-03-2025

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Finding the least common multiple (LCM) is a fundamental concept in mathematics, particularly useful in simplifying fractions and solving problems involving cycles or repeating events. This article will clearly explain how to find the LCM of 6 and 3, and will also explore different methods to solve similar problems.

Understanding Least Common Multiple

The least common multiple (LCM) of two or more numbers is the smallest positive integer that is a multiple of all the numbers. In simpler terms, it's the smallest number that all the given numbers can divide into evenly.

For example, let's consider the numbers 2 and 3. Multiples of 2 are 2, 4, 6, 8, 10, and so on. Multiples of 3 are 3, 6, 9, 12, 15, and so on. The smallest number that appears in both lists is 6. Therefore, the LCM of 2 and 3 is 6.

Finding the LCM of 6 and 3

Now, let's find the LCM of 6 and 3. We can use a few different methods:

Method 1: Listing Multiples

• Multiples of 6: 6, 12, 18, 24, 30...
• Multiples of 3: 3, 6, 9, 12, 15...

Notice that the smallest number that appears in both lists is 6. Therefore, the LCM of 6 and 3 is $\boxed{6}$.

Method 2: Prime Factorization

This method is particularly useful for larger numbers.

1. Find the prime factorization of each number:

   • 6 = 2 x 3
   • 3 = 3

2. Identify the highest power of each prime factor:

   • The prime factors are 2 and 3.
   • The highest power of 2 is 2¹ = 2.
   • The highest power of 3 is 3¹ = 3.

3. Multiply the highest powers together:

   • LCM(6, 3) = 2 x 3 = 6

Therefore, the LCM of 6 and 3 is $\boxed{6}$.

Method 3: Using the Greatest Common Divisor (GCD)

There's a relationship between the LCM and the greatest common divisor (GCD) of two numbers:

LCM(a, b) = (|a * b|) / GCD(a, b)

1. Find the GCD of 6 and 3: The greatest common divisor of 6 and 3 is 3 (because 3 is the largest number that divides both 6 and 3 evenly).

2. Apply the formula: LCM(6, 3) = (6 * 3) / 3 = 6

Therefore, the LCM of 6 and 3 is $\boxed{6}$.

Why is the LCM Important?

The LCM has practical applications in various areas:

• Fractions: Finding a common denominator when adding or subtracting fractions.
• Scheduling: Determining when events with different repeating cycles will occur simultaneously. For example, if one event happens every 6 days and another every 3 days, they'll both occur on the 6th day.
• Modular Arithmetic: Solving problems related to remainders and cycles.

Understanding the LCM helps in solving a wide range of mathematical problems efficiently. The methods outlined above provide different approaches to finding the LCM, allowing you to choose the most suitable method depending on the numbers involved.