Least Common Multiple 5 6 7

Finding the Least Common Multiple (LCM) of 5, 6, and 7

This article will guide you through calculating the least common multiple (LCM) of 5, 6, and 7. The LCM is the smallest positive integer that is a multiple of all the given numbers. Understanding LCM is crucial in various mathematical applications, from simplifying fractions to solving problems involving cycles and timing. This guide will explain the concept and provide a step-by-step solution for finding the LCM of 5, 6, and 7.

Understanding Least Common Multiple (LCM)

The least common multiple, or LCM, is the smallest number that is a multiple of two or more numbers. Think of it as the smallest number that all the given numbers can divide into evenly. It’s different from the greatest common divisor (GCD), which is the largest number that divides all the given numbers without leaving a remainder.

Methods for Finding the LCM

There are several ways to find the LCM, and the best method depends on the numbers involved. For smaller numbers like 5, 6, and 7, we can use a couple of approaches:

1. Listing Multiples

This method is straightforward, especially for smaller numbers. List the multiples of each number until you find the smallest multiple common to all.

• Multiples of 5: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, ...
• Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, ...
• Multiples of 7: 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, ...

By examining these lists, we can see that the smallest multiple common to 5, 6, and 7 is 210. Therefore, the LCM(5, 6, 7) = 210.

2. Prime Factorization Method

This method is more efficient for larger numbers or a greater number of integers. It involves finding the prime factorization of each number and then constructing the LCM from the highest powers of each prime factor.

• Prime factorization of 5: 5 (5 is a prime number)
• Prime factorization of 6: 2 x 3
• Prime factorization of 7: 7 (7 is a prime number)

To find the LCM, we take the highest power of each prime factor present in the factorizations:

• Highest power of 2: 2¹ = 2
• Highest power of 3: 3¹ = 3
• Highest power of 5: 5¹ = 5
• Highest power of 7: 7¹ = 7

Multiply these highest powers together: 2 x 3 x 5 x 7 = 210. Therefore, the LCM(5, 6, 7) = 210.

Conclusion:

The least common multiple of 5, 6, and 7 is 210. Both the listing multiples and prime factorization methods lead to the same result. The prime factorization method is generally preferred for larger numbers as it is a more systematic and efficient approach. Understanding LCM is essential for various mathematical operations and problem-solving scenarios. Now you have the tools to tackle similar LCM problems with confidence!