what is the least common multiple of 6 and 7

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30-03-2025

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Finding the least common multiple (LCM) is a fundamental concept in mathematics, particularly in arithmetic and algebra. Understanding LCMs helps in simplifying fractions, solving equations, and understanding rhythmic patterns. This article will clearly explain how to find the LCM of 6 and 7.

Understanding Least Common Multiple

The least common multiple (LCM) of two or more numbers is the smallest positive integer that is divisible by all the numbers. For example, the LCM of 2 and 3 is 6 because 6 is the smallest number divisible by both 2 and 3.

Methods for Finding the LCM

There are several ways to calculate the LCM. Let's explore two common methods to find the LCM of 6 and 7:

Method 1: Listing Multiples

This method involves listing the multiples of each number until you find the smallest multiple common to both.

• Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48...
• Multiples of 7: 7, 14, 21, 28, 35, 42, 49...

Notice that 42 is the smallest multiple present in both lists. Therefore, the LCM of 6 and 7 is 42.

Method 2: Prime Factorization

This method uses the prime factorization of each number. Prime factorization is expressing a number as a product of its prime factors.

1. Find the prime factorization of each number:

   • 6 = 2 x 3
   • 7 = 7 (7 is a prime number)

2. Identify the highest power of each prime factor:

   • The prime factors are 2, 3, and 7. The highest power of 2 is 2¹, the highest power of 3 is 3¹, and the highest power of 7 is 7¹.

3. Multiply the highest powers together:

   • LCM(6, 7) = 2¹ x 3¹ x 7¹ = 42

Therefore, using prime factorization, we again find that the LCM of 6 and 7 is 42.

Why is the LCM of 6 and 7, 42?

The LCM of 6 and 7 is 42 because 42 is the smallest positive integer that is divisible by both 6 and 7 without leaving a remainder. No smaller number satisfies this condition.

Applications of LCM

Understanding LCMs is crucial in various mathematical applications, including:

• Fraction addition and subtraction: Finding a common denominator when adding or subtracting fractions.
• Solving problems involving cycles or repeating events: Determining when events will coincide. For example, if two events repeat every 6 and 7 days respectively, they will coincide again after 42 days.
• Scheduling and planning: Coordinating tasks or events that occur at different intervals.

Conclusion

The least common multiple of 6 and 7 is 42. Both the listing multiples method and the prime factorization method effectively demonstrate this. Understanding LCMs is a fundamental skill with practical applications in various areas of mathematics and beyond. Remember to choose the method you find easiest and most efficient for your calculations.