Least Common Multiple Of 28 And 35

Finding the Least Common Multiple (LCM) of 28 and 35

This article will guide you through understanding and calculating the least common multiple (LCM) of 28 and 35. We'll explore different methods, making this concept clear and accessible for everyone, from students learning about multiples to those needing a refresher. Understanding LCM is crucial in various mathematical applications, including fractions, scheduling problems, and more.

What is the Least Common Multiple (LCM)?

The least common multiple (LCM) of two or more numbers is the smallest positive integer that is a multiple of each of the numbers. In simpler terms, it's the smallest number that both numbers can divide into evenly. For example, the LCM of 2 and 3 is 6, because 6 is the smallest number that is divisible by both 2 and 3.

Methods for Finding the LCM of 28 and 35

There are several ways to find the LCM of 28 and 35. Let's explore two common approaches:

1. Listing Multiples Method

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

• Multiples of 28: 28, 56, 84, 112, 140, 168, ...
• Multiples of 35: 35, 70, 140, 175, ...

As you can see, the smallest common multiple of 28 and 35 is 140. This method works well for smaller numbers, but it can become time-consuming for larger numbers.

2. Prime Factorization Method

This method is generally more efficient, especially for larger numbers. It involves finding the prime factorization of each number and then constructing the LCM from the prime factors.

• Prime factorization of 28: 2 x 2 x 7 = 2² x 7
• Prime factorization of 35: 5 x 7

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

• The highest power of 2 is 2² = 4
• The highest power of 5 is 5¹ = 5
• The highest power of 7 is 7¹ = 7

Therefore, the LCM of 28 and 35 is 2² x 5 x 7 = 4 x 5 x 7 = 140.

3. Greatest Common Divisor (GCD) Method

This method uses the relationship between the LCM and the Greatest Common Divisor (GCD) of two numbers. The formula is:

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

First, we find the GCD of 28 and 35 using the Euclidean algorithm or prime factorization:

• Prime factorization of 28: 2² x 7
• Prime factorization of 35: 5 x 7

The common prime factor is 7, therefore the GCD(28, 35) = 7.

Now, apply the formula:

LCM(28, 35) = (28 x 35) / 7 = 980 / 7 = 140

Conclusion:

Regardless of the method used, the least common multiple of 28 and 35 is 140. Choosing the most efficient method depends on the numbers involved. For smaller numbers, listing multiples might be quicker. For larger numbers, prime factorization or the GCD method is generally more efficient and less prone to errors. Understanding LCM is a fundamental skill with broad applications in mathematics and beyond.