Understanding the Relationship Between Sequence Numbers, LCM, and GCF

The relationship between sequence numbers, Least Common Multiple (LCM), and Greatest Common Factor (GCF) is a fundamental concept in mathematics, particularly in number theory. In this article, we will explore a specific scenario involving sequence numbers in the ratio of 4:5:6, their LCM, and GCF, and provide a detailed explanation of how to find these values.

Scenario Introduction

Consider three numbers in the ratio 4:5:6. Their LCM is 600. The goal is to find the Greatest Common Factor (GCF) of these numbers. Let's break this down step by step.

Step-by-Step Solution

Given:
The numbers are in the ratio 4:5:6.
LCM of the three numbers is 600.

Method 1: Direct Calculation

Using the given information, we start by considering the three numbers as 4k, 5k, and 6k, where k is the GCF of these numbers. According to the properties of LCM and GCF:

Step 1: Express the LCM in terms of k:

The LCM of 4k, 5k, and 6k is 60k.

Step 2: Set up the equation with the given LCM:

60k 600

Step 3: Solve for k:

k 600 / 60 10

Step 4: Determine the three numbers:

The three numbers are 40, 50, and 60.

Step 5: Calculate the GCF:

The GCF of 40, 50, and 60 is calculated by finding the prime factors:

40 23 × 5
50 2 × 52
60 22 × 3 × 5

The GCF is the product of the lowest powers of the common prime factors:

21 × 51 10

Therefore, the GCF is 10.

Method 2: Verification

To verify the solution, we can use the prime factorization method and the definitions of LCM and GCF:

Step 1: Prime factorize the numbers:

40 23 × 5
50 2 × 52
60 22 × 3 × 5

Step 2: Calculate the LCM:

LCM 23 × 3 × 52 600

Step 3: Calculate the GCF:

GCF 21 × 51 10

This confirms our solution.

Additional Verification

Another way to verify the numbers and their GCF is through a direct check:

Step 1: Calculate the LCM:

LCM of 4, 5, 6 60

Step 2: Multiply by k (the common factor):

60k 600

Step 3: Solve for k:

k 10

Step 4: The numbers are 40, 50, and 60.

Step 5: Confirm the GCF:

GCF 10

This method also verifies our solution.

In conclusion, the GCF of the three numbers in the ratio 4:5:6 with an LCM of 600 is 10. This problem highlights the importance of understanding the relationships between ratios, LCM, and GCF in solving mathematical problems.