Finding The Other Number GCD 2 LCM 180 And One Number 18

Jun 27, 2025 by ADMIN 57 views

If the GCD of two numbers is 2, the LCM is 180, and one number is 18, what is the other number?

Finding the Other Number When GCD is 2 and LCM is 180, and One Number is 18

In mathematics, understanding the relationship between the greatest common divisor (GCD) and the least common multiple (LCM) is crucial for solving various number theory problems. This article delves into a specific problem where we are given the GCD and LCM of two numbers, along with one of the numbers, and are tasked with finding the other number. We will explore the underlying principles and apply them to solve the problem effectively. Specifically, we'll address the question: If the GCD of two numbers is 2, the LCM is 180, and one of the numbers is 18, what is the other number?

Understanding GCD and LCM

Before diving into the solution, let's briefly recap the concepts of GCD and LCM. The GCD of two or more numbers is the largest positive integer that divides each of the numbers without leaving a remainder. For instance, the GCD of 12 and 18 is 6, as 6 is the largest number that divides both 12 and 18. The LCM, on the other hand, is the smallest positive integer that is divisible by each of the numbers. For example, the LCM of 12 and 18 is 36, as 36 is the smallest number that both 12 and 18 divide into evenly.

The relationship between GCD and LCM is fundamental in number theory. For any two positive integers, say a and b, the product of their GCD and LCM is equal to the product of the numbers themselves. Mathematically, this can be expressed as:

GCD(a, b) * LCM(a, b) = a * b

This formula is a cornerstone for solving problems involving GCD and LCM, and we will utilize it to solve the problem at hand.

Problem Statement

We are given the following information:

The GCD of two numbers is 2.
The LCM of the two numbers is 180.
One of the numbers is 18.

Our goal is to find the other number. Let's denote the two numbers as a and b, where a = 18. We need to find the value of b.

Applying the GCD and LCM Relationship

Using the formula GCD(a, b) * LCM(a, b) = a * b, we can substitute the given values:

2 * 180 = 18 * b

Now, we need to solve for b.

Solving for the Unknown Number

Let's simplify the equation:

360 = 18 * b

To isolate b, we divide both sides of the equation by 18:

b = 360 / 18

b = 20

Therefore, the other number is 20.

Verification

To ensure our solution is correct, let's verify if the GCD of 18 and 20 is indeed 2, and their LCM is 180.

Finding the GCD of 18 and 20

The factors of 18 are: 1, 2, 3, 6, 9, 18 The factors of 20 are: 1, 2, 4, 5, 10, 20

The common factors of 18 and 20 are 1 and 2. The greatest of these is 2. Thus, GCD(18, 20) = 2.

Finding the LCM of 18 and 20

One way to find the LCM is to list the multiples of each number until we find a common multiple:

Multiples of 18: 18, 36, 54, 72, 90, 108, 126, 144, 162, 180, ... Multiples of 20: 20, 40, 60, 80, 100, 120, 140, 160, 180, ...

The smallest common multiple is 180. Thus, LCM(18, 20) = 180.

Since the GCD of 18 and 20 is 2, and their LCM is 180, our solution is verified.

Conclusion

In conclusion, when given the GCD and LCM of two numbers, along with one of the numbers, we can find the other number by utilizing the relationship GCD(a, b) * LCM(a, b) = a * b. By substituting the given values and solving for the unknown, we determined that the other number is 20. We further verified our solution by calculating the GCD and LCM of 18 and 20, confirming that they match the given values. Understanding and applying these concepts is essential for tackling number theory problems effectively. This problem illustrates how fundamental mathematical principles can be used to solve practical problems, emphasizing the importance of a strong foundation in mathematical concepts.

Understanding number theory concepts like the greatest common divisor (GCD) and the least common multiple (LCM) is essential for solving various mathematical problems. This article addresses a specific problem where we need to find an unknown number given the GCD and LCM of two numbers, along with one of the numbers. The problem we will solve is: If the GCD of two numbers is 2, the LCM is 180, and one of the numbers is 18, what is the other number?

The Significance of GCD and LCM

The GCD, also known as the greatest common factor (GCF), is the largest positive integer that divides two or more numbers without any remainder. For example, the GCD of 24 and 36 is 12, because 12 is the largest number that divides both 24 and 36 perfectly. The LCM, conversely, is the smallest positive integer that is divisible by each of the given numbers. For instance, the LCM of 24 and 36 is 72, as 72 is the smallest number that both 24 and 36 can divide into evenly.

The relationship between the GCD and LCM is a critical concept in number theory. For any two positive integers a and b, the product of their GCD and LCM is equal to the product of the numbers themselves. This relationship is expressed by the formula:

GCD(a, b) * LCM(a, b) = a * b

This formula serves as the backbone for solving a variety of problems involving GCD and LCM. It provides a direct link between these two concepts and the original numbers, allowing us to find missing values when others are known. In our problem, we will utilize this formula to determine the value of the unknown number.

Problematic Details

In the problem presented, we have the following information:

The GCD of two numbers is 2.
The LCM of the two numbers is 180.
One of the numbers is 18.

Our task is to find the second number. Let's denote the two numbers as a and b, where a is given as 18. We need to calculate the value of b.

Applying the Core Formula

Using the formula GCD(a, b) * LCM(a, b) = a * b, we substitute the provided values into the equation:

2 * 180 = 18 * b

This equation now allows us to solve for the unknown variable b.

Calculation Process

Let's simplify the equation to find the value of b:

360 = 18 * b

To isolate b, we divide both sides of the equation by 18:

b = 360 / 18

b = 20

Thus, the other number is 20.

Verification Steps

To ensure the accuracy of our solution, we must verify that the GCD of 18 and 20 is indeed 2, and their LCM is 180.

GCD Calculation for 18 and 20

List the factors of 18: 1, 2, 3, 6, 9, 18 List the factors of 20: 1, 2, 4, 5, 10, 20

The common factors of 18 and 20 are 1 and 2. The greatest common factor is 2. Therefore, GCD(18, 20) = 2.

LCM Calculation for 18 and 20

We can find the LCM by listing the multiples of each number until we find the smallest common multiple:

Multiples of 18: 18, 36, 54, 72, 90, 108, 126, 144, 162, 180, ... Multiples of 20: 20, 40, 60, 80, 100, 120, 140, 160, 180, ...

The smallest common multiple is 180. Thus, LCM(18, 20) = 180.

Since the GCD of 18 and 20 is 2, and their LCM is 180, our calculated value for the other number is correct.

Final Thoughts

In summary, we successfully found the other number by applying the relationship between GCD, LCM, and the numbers themselves. Given that the GCD of two numbers is 2, their LCM is 180, and one of the numbers is 18, we determined that the other number is 20. This was achieved by utilizing the formula GCD(a, b) * LCM(a, b) = a * b and verifying our result through GCD and LCM calculations. This problem highlights the importance of understanding fundamental number theory concepts and their applications in solving mathematical problems. Mastering these concepts allows for a deeper appreciation of mathematical relationships and enhances problem-solving skills.

In the realm of number theory, the concepts of the greatest common divisor (GCD) and the least common multiple (LCM) are pivotal. They provide a framework for understanding the relationships between numbers and are fundamental in solving various mathematical problems. This article will focus on solving a specific problem that involves finding an unknown number when the GCD and LCM of two numbers, along with one of the numbers, are provided. Specifically, we aim to answer the question: If the GCD of two numbers is 2, their LCM is 180, and one of the numbers is 18, what is the other number?

Delving into GCD and LCM

The GCD, also known as the greatest common factor (GCF), is the largest positive integer that perfectly divides two or more numbers without leaving any remainder. For example, the GCD of 30 and 45 is 15 because 15 is the largest number that divides both 30 and 45. Conversely, the LCM is the smallest positive integer that is divisible by each of the given numbers. For instance, the LCM of 30 and 45 is 90 because 90 is the smallest number that both 30 and 45 divide into evenly.

The relationship between the GCD and LCM is a cornerstone of number theory. For any two positive integers a and b, the product of their GCD and their LCM is equivalent to the product of the two numbers themselves. This relationship is mathematically represented as:

GCD(a, b) * LCM(a, b) = a * b

This formula is not just a theoretical construct; it is a powerful tool for solving a wide range of problems involving GCD and LCM. It establishes a direct link between these two concepts and the original numbers, making it possible to find missing values when some information is known. We will apply this formula to solve the specific problem at hand.

Laying Out the Specifics

The problem statement provides us with the following information:

The GCD of two numbers is 2.
The LCM of the same two numbers is 180.
One of the numbers is 18.

Our objective is to find the value of the other number. Let's denote the two numbers as a and b, where a is given as 18. We need to determine the value of b.

The Application of the Formula

By employing the fundamental formula GCD(a, b) * LCM(a, b) = a * b, we can substitute the known values into the equation:

2 * 180 = 18 * b

This equation now sets the stage for solving for the unknown value, b.

Step-by-Step Solution

Let's proceed to simplify the equation and isolate b:

360 = 18 * b

To find the value of b, we divide both sides of the equation by 18:

b = 360 / 18

b = 20

Therefore, the other number is 20.

Confirmation of the Result

To ensure the correctness of our solution, it is essential to verify that the GCD of 18 and 20 is indeed 2, and their LCM is 180.

GCD Verification for 18 and 20

Listing the factors of 18: 1, 2, 3, 6, 9, 18 Listing the factors of 20: 1, 2, 4, 5, 10, 20

The common factors of 18 and 20 are 1 and 2. The greatest among these is 2. Hence, GCD(18, 20) = 2.

LCM Verification for 18 and 20

We can find the LCM by listing the multiples of each number until a common multiple is identified:

Multiples of 18: 18, 36, 54, 72, 90, 108, 126, 144, 162, 180, ... Multiples of 20: 20, 40, 60, 80, 100, 120, 140, 160, 180, ...

The smallest multiple that is common to both lists is 180. Thus, LCM(18, 20) = 180.

Since the GCD of 18 and 20 is 2, and their LCM is 180, our calculated value for the second number is accurate.

Final Conclusion

In conclusion, we have successfully determined the other number by leveraging the relationship between GCD, LCM, and the numbers themselves. Given that the GCD of two numbers is 2, their LCM is 180, and one of the numbers is 18, we found that the second number is 20. This was achieved by applying the formula GCD(a, b) * LCM(a, b) = a * b and subsequently verifying our result through the calculation of GCD and LCM. This problem exemplifies the significance of understanding and applying fundamental number theory principles to solve mathematical problems. A solid grasp of these concepts not only enhances problem-solving capabilities but also fosters a deeper appreciation for the elegance and interconnectedness of mathematical ideas.