Calculating LCM of 105 and 189

Oct 13, 2025 by miodeya.com 33 views

Introduction

In mathematics, the Least Common Multiple (LCM) is a fundamental concept that plays a crucial role in various mathematical operations. It is the smallest multiple that is common to two or more numbers. In this article, we will delve into the concept of LCM and calculate the LCM of 105 and 189.

What is LCM?

The LCM of two numbers is the smallest number that is a multiple of both numbers. It is also known as the smallest common multiple (SCM). The LCM is an essential concept in mathematics, particularly in number theory, algebra, and geometry.

How to Calculate LCM

To calculate the LCM of two numbers, we can use the following methods:

Method 1: Listing Multiples

This method involves listing the multiples of each number and finding the smallest common multiple.

Multiples of 105: 105, 210, 315, 420, 525, 630, 735, 840, 945, 1050, ...
Multiples of 189: 189, 378, 567, 756, 945, 1134, 1323, 1512, 1701, 1890, ...

From the list, we can see that the smallest common multiple of 105 and 189 is 945.

Method 2: Prime Factorization

This method involves finding the prime factorization of each number and then multiplying the highest power of each prime factor.

Prime factorization of 105: 3 × 5 × 7
Prime factorization of 189: 3 × 3 × 3 × 7

To find the LCM, we multiply the highest power of each prime factor:

LCM = 3^3 × 5 × 7 = 945

Calculating LCM of 105 and 189

Using the above methods, we can calculate the LCM of 105 and 189.

Method 1: Listing Multiples
- Multiples of 105: 105, 210, 315, 420, 525, 630, 735, 840, 945, 1050, ...
- Multiples of 189: 189, 378, 567, 756, 945, 1134, 1323, 1512, 1701, 1890, ...
- Smallest common multiple: 945
Method 2: Prime Factorization
- Prime factorization of 105: 3 × 5 × 7
- Prime factorization of 189: 3 × 3 × 3 × 7
- LCM = 3^3 × 5 × 7 = 945

Conclusion

In conclusion, the LCM of 105 and 189 is 945. We have used two methods to calculate the LCM: listing multiples and prime factorization. Both methods have yielded the same result, confirming that the LCM of 105 and 189 is indeed 945.

Frequently Asked Questions

Q: What is the LCM of 105 and 189?

A: The LCM of 105 and 189 is 945.

Q: How do I calculate the LCM of two numbers?

A: You can use either the listing multiples method or the prime factorization method to calculate the LCM.

Q: What is the significance of LCM in mathematics?

A: LCM is an essential concept in mathematics, particularly in number theory, algebra, and geometry. It plays a crucial role in various mathematical operations and is used to solve problems in these fields.

References

[1] Khan Academy. (n.d.). Least Common Multiple (LCM). Retrieved from https://www.khanacademy.org/math/algebra/x2-2-advanced-algebra-topic/x2-2-2-advanced-algebra-topic-2/v/least-common-multiple-lcm
[2] Math Open Reference. (n.d.). Least Common Multiple (LCM). Retrieved from https://www.mathopenref.com/lcm.html

Q&A: Frequently Asked Questions

Q: What is the LCM of 105 and 189?

A: The LCM of 105 and 189 is 945.

Q: How do I calculate the LCM of two numbers?

A: You can use either the listing multiples method or the prime factorization method to calculate the LCM.

Listing Multiples Method

To calculate the LCM using the listing multiples method, follow these steps:

List the multiples of each number.
Find the smallest common multiple.

Example:

Multiples of 105: 105, 210, 315, 420, 525, 630, 735, 840, 945, 1050, ...
Multiples of 189: 189, 378, 567, 756, 945, 1134, 1323, 1512, 1701, 1890, ...

From the list, we can see that the smallest common multiple of 105 and 189 is 945.

Prime Factorization Method

To calculate the LCM using the prime factorization method, follow these steps:

Find the prime factorization of each number.
Multiply the highest power of each prime factor.

Example:

Prime factorization of 105: 3 × 5 × 7
Prime factorization of 189: 3 × 3 × 3 × 7

To find the LCM, we multiply the highest power of each prime factor:

LCM = 3^3 × 5 × 7 = 945

Q: What is the significance of LCM in mathematics?

Q: How do I use LCM in real-life situations?

A: LCM is used in various real-life situations, such as:

Music: LCM is used to find the lowest common multiple of two or more notes to create a harmonious sound.
Cooking: LCM is used to find the lowest common multiple of two or more ingredients to create a recipe.
Engineering: LCM is used to find the lowest common multiple of two or more components to create a design.

Q: Can I use LCM to solve problems in other areas of mathematics?

A: Yes, LCM can be used to solve problems in other areas of mathematics, such as:

Algebra: LCM is used to solve equations and inequalities.
Geometry: LCM is used to find the area and perimeter of shapes.
Number Theory: LCM is used to find the prime factorization of numbers.

Q: How do I find the LCM of three or more numbers?

A: To find the LCM of three or more numbers, follow these steps:

Find the LCM of two numbers.
Find the LCM of the result and the third number.

Example:

LCM of 105 and 189: 945
LCM of 945 and 315: 945 × 3 = 2835

Q: Can I use a calculator to find the LCM?

A: Yes, you can use a calculator to find the LCM. Most calculators have a built-in function to find the LCM.

Q: How do I check if my answer is correct?

A: To check if your answer is correct, follow these steps:

List the multiples of each number.
Find the smallest common multiple.

If your answer matches the smallest common multiple, then it is correct.

Conclusion

Frequently Asked Questions

Q: What is the LCM of 105 and 189?

A: The LCM of 105 and 189 is 945.

Q: How do I calculate the LCM of two numbers?

A: You can use either the listing multiples method or the prime factorization method to calculate the LCM.

Q: What is the significance of LCM in mathematics?

References

[1] Khan Academy. (n.d.). Least Common Multiple (LCM). Retrieved from https://www.khanacademy.org/math/algebra/x2-2-advanced-algebra-topic/x2-2-2-advanced-algebra-topic-2/v/least-common-multiple-lcm
[2] Math Open Reference. (n.d.). Least Common Multiple (LCM). Retrieved from https://www.mathopenref.com/lcm.html