what is the least common multiple (lcm) of 7 and 8?

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30-03-2025

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Finding the least common multiple (LCM) is a fundamental concept in math. This article will clearly explain how to calculate the LCM of 7 and 8, and provide you with a few different methods to solve this and similar problems. Understanding LCMs is crucial for various mathematical operations, from simplifying fractions to solving more complex algebraic equations. So, let's dive in and find the LCM of 7 and 8!

Understanding Least Common Multiple (LCM)

Before we tackle the specific problem, let's define what the least common multiple actually is. The LCM of two or more numbers is the smallest positive integer that is divisible by all the numbers. Think of it as the smallest number that all the numbers in question can divide into evenly.

Method 1: Listing Multiples

One straightforward method to find the LCM is to list the multiples of each number until you find the smallest common multiple.

• Multiples of 7: 7, 14, 21, 28, 35, 42, 49, 56, 63, 70...
• Multiples of 8: 8, 16, 24, 32, 40, 48, 56, 64, 72, 80...

Notice that 56 appears in both lists. Therefore, the least common multiple of 7 and 8 is 56.

Method 2: Prime Factorization

This method is particularly useful for larger numbers. We break down each number into its prime factors.

• Prime factorization of 7: 7 (7 is a prime number)
• Prime factorization of 8: 2 x 2 x 2 = 2³

To find the LCM using prime factorization, we take the highest power of each prime factor present in either factorization and multiply them together. In this case:

2³ x 7 = 8 x 7 = 56

Again, the LCM of 7 and 8 is 56.

Method 3: Formula using Greatest Common Divisor (GCD)

There's a handy formula that relates the LCM and the greatest common divisor (GCD) of two numbers:

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

Where:

• a and b are the two numbers.
• GCD(a, b) is the greatest common divisor of a and b.

Since 7 and 8 have no common factors other than 1, their GCD is 1. Therefore:

LCM(7, 8) = (7 x 8) / 1 = 56

Once more, the LCM of 7 and 8 is 56.

Why is the LCM Important?

Understanding LCM is crucial for various mathematical applications, including:

• Fraction addition and subtraction: Finding a common denominator is essential for adding or subtracting fractions, and the LCM provides the least common denominator.
• Solving word problems: Many real-world problems involving cycles or repeating events can be solved using the LCM. For example, determining when two events will occur simultaneously.
• Simplifying expressions: The LCM can help simplify complex algebraic expressions.

Conclusion

We've explored three different methods to determine the least common multiple of 7 and 8. No matter which method you use, you'll arrive at the same answer: 56. Understanding the concept of LCM and mastering these methods will be invaluable in your mathematical journey. Remember, practice makes perfect! Try finding the LCM of other number pairs to solidify your understanding.