Common Multiple and Least Common Multiple: A Comprehensive Guide

Introduction

Understanding the concepts of common multiple and least common multiple (LCM) is fundamental in solving various mathematical problems, particularly in contexts like the meeting of runners on a circular track. This article will explore these concepts through detailed examples and explain how LCM is applied in real-world scenarios.

What is a Common Multiple?

A common multiple of two or more numbers is a number that is a multiple of each of the numbers. For example, the common multiples of 2, 4, and 5.5 are numbers like 88, 176, and so on. This concept is particularly useful in situations where evenly spaced intervals or periodic events need to align.

Least Common Multiple (LCM)

The least common multiple is the smallest positive integer that is divisible by each of the numbers. In mathematical terms, it is the smallest positive integer that is a common multiple of two or more numbers. LCM is widely used in various fields, including mathematics, engineering, and even in optimizing computer algorithms.

Calculating LCM Using Fractions

Let's consider the problem of three runners running around a circular track. If the times to complete one revolution are 2 hours, 4 hours, and 5.5 hours, we need to find when the runners will all meet at the starting point. Here's a step-by-step guide on how to do this:

Step 1: Express Times in a Common Format

First, we express each time in terms of fractions:

Runner 1: 2 (frac{2}{1}) Runner 2: 4 (frac{4}{1}) Runner 3: 5.5 (frac{11}{2})

Step 2: Find the LCM of the Denominators

The denominators are 1, 1, and 2. The least common multiple of these denominators is 2.

Step 3: Convert to a Common Denominator

Next, we convert each time to have the common denominator of 2:

Runner 1: (frac{2}{1} frac{4}{2}) Runner 2: (frac{4}{1} frac{8}{2}) Runner 3: (frac{11}{2})

Step 4: Find the LCM of the Numerators

Now, we find the LCM of the numerators: 4, 8, and 11. The prime factorizations are:

4 (2^2) 8 (2^3) 11 11 (a prime number)

The LCM takes the highest power of each prime:

For 2, the highest power is (2^3) For 11, the highest power is 11^1

Thus, the LCM is: [text{LCM} 2^3 times 11 8 times 11 88]

Step 5: Convert Back to Hours

Since we had a common denominator of 2, we need to convert the LCM back to hours:

[text{LCM in hours} frac{88}{2} 44 text{ hours}]

Conclusion: The three runners will meet at the starting point again after 44 hours.

Additional Examples

Let's look at a couple of additional examples to solidify our understanding of LCM:

LCM of 23 and 4 is 12

In this case, the three runners meet again at the starting point after 12 hours. This is because 12 is the smallest number that is a multiple of both 23 and 4.

Times in decimal and fraction form: 2, 4, 5.5

Again, we convert all the numbers into fractions: 2/1, 4/1, 11/2. Using the LCM method, the numerators are 2411, and the denominators are 112. The LCM of the numerators is 44, and the HCF of the denominators is 1, so the LCM is 44.

Multiplying times by 10: 4, 6, 7.5

Multiply 4, 6, 7.5 by 10: 40, 60, 75. Factorizing, we get:

40 (2^3 times 5) 60 (2^2 times 3 times 5) 75 (3 times 5^2)

The LCM of 40, 60, 7.5 is 600, so the LCM of 4, 6, 7.5 is 600/10 60. Therefore, the runners will meet after 60 hours after starting the race.

Conclusion

The application of LCM in real-world scenarios, such as the meeting of runners on a circular track, demonstrates its importance in solving practical problems. Whether in fractions or decimals, understanding LCM allows for precise scheduling and timing in various scenarios.