Least Common Multiple Of 5 6 7

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

This article will guide you through the process of calculating the least common multiple (LCM) of 5, 6, and 7. Understanding LCM is crucial in various mathematical applications, from simplifying fractions to solving problems involving cyclical events. We'll explore different methods, making this concept accessible to everyone, from beginners to those looking for a refresher.

What is the Least Common Multiple (LCM)?

The least common multiple (LCM) is the smallest positive integer that is divisible by all the numbers in a given set. In simpler terms, it's the smallest number that all the numbers in the set can divide into evenly without leaving a remainder. 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 of 5, 6, and 7

We'll explore two primary methods to find the LCM of 5, 6, and 7: the prime factorization method and the listing multiples method.

1. Prime Factorization Method:

This method involves breaking down each number into its prime factors. Prime factors are numbers that are only divisible by 1 and themselves (e.g., 2, 3, 5, 7, 11...).

• Step 1: Prime Factorization:

  • 5 = 5
  • 6 = 2 x 3
  • 7 = 7

• Step 2: Identify the highest power of each prime factor:

  • The prime factors present are 2, 3, 5, and 7.
  • The highest power of 2 is 2¹ = 2
  • The highest power of 3 is 3¹ = 3
  • The highest power of 5 is 5¹ = 5
  • The highest power of 7 is 7¹ = 7

• Step 3: Multiply the highest powers together:

  • LCM(5, 6, 7) = 2 x 3 x 5 x 7 = 210

Therefore, the least common multiple of 5, 6, and 7 is 210.

2. Listing Multiples Method:

This method involves listing out the multiples of each number until you find the smallest multiple common to all. While effective for smaller numbers, it becomes less efficient with larger numbers.

• Multiples of 5: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 100, 105, 110, 115, 120, 125, 130, 135, 140, 145, 150, 155, 160, 165, 170, 175, 180, 185, 190, 195, 200, 205, 210,...
• Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, 78, 84, 90, 96, 102, 108, 114, 120, 126, 132, 138, 144, 150, 156, 162, 168, 174, 180, 186, 192, 198, 204, 210,...
• Multiples of 7: 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84, 91, 98, 105, 112, 119, 126, 133, 140, 147, 154, 161, 168, 175, 182, 189, 196, 203, 210,...

The smallest multiple common to all three lists is 210.

Conclusion:

Both methods confirm that the least common multiple of 5, 6, and 7 is 210. The prime factorization method is generally more efficient for larger numbers, while the listing multiples method is easier to grasp conceptually for beginners. Understanding how to calculate the LCM is a valuable skill in various mathematical contexts. Remember to practice both methods to solidify your understanding.