What Is The Lcm For 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 cycles and timing. This simple example will help you grasp the core concept and the different methods used to find the LCM.

What is the Least Common Multiple (LCM)?

The Least Common Multiple (LCM) is the smallest positive integer that is a multiple of two or more integers. In simpler terms, it's the smallest number that all the given numbers can divide into evenly. For instance, the LCM of 2 and 3 is 6 because 6 is the smallest number divisible by both 2 and 3.

Method 1: Prime Factorization

This is a widely used and efficient method for finding the LCM, especially when dealing with larger numbers. Here's how it works for 5, 6, and 7:

1. Find the prime factorization of each number:

   • 5 = 5 (5 is a prime number)
   • 6 = 2 x 3 (2 and 3 are prime numbers)
   • 7 = 7 (7 is a prime number)

2. Identify the highest power of each prime factor:

   • The prime factors 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

3. Multiply the highest powers together:

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

Therefore, the LCM of 5, 6, and 7 is 210.

Method 2: Listing Multiples

This method is suitable for smaller numbers. It involves listing the multiples of each number until you find the smallest common multiple.

1. List 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...

2. List 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...

3. List 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 number 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 method is useful for a quick understanding of the concept with smaller numbers. Remember that understanding LCM is essential for various mathematical operations and problem-solving.