Find the prime factors of each number

Oct 13, 2025 by talynt.org 28 views

Introduction

In mathematics, the Least Common Multiple (LCM) and the Highest Common Factor (HCF) are two fundamental concepts that play a crucial role in number theory. The LCM of a set of numbers is the smallest number that is a multiple of each of the numbers in the set, while the HCF is the largest number that divides each of the numbers in the set without leaving a remainder. In this article, we will explore how to determine the LCM and HCF of three given numbers using prime factors.

What are Prime Factors?

Prime factors are the prime numbers that multiply together to give a specific number. For example, the prime factors of 12 are 2 and 3, since 2 × 2 × 3 = 12. Prime factors are essential in determining the LCM and HCF of a set of numbers.

Methodology

To determine the LCM and HCF of three given numbers, we will follow these steps:

Find the prime factors of each number: We will start by finding the prime factors of each of the three given numbers.
Identify the common prime factors: We will then identify the common prime factors among the three numbers.
Determine the LCM: We will use the common prime factors to determine the LCM of the three numbers.
Determine the HCF: We will use the common prime factors to determine the HCF of the three numbers.

Step 1: Find the Prime Factors of Each Number

Let's start by finding the prime factors of each of the three given numbers.

Number 1: 848

To find the prime factors of 848, we will start by dividing it by the smallest prime number, which is 2.

848 ÷ 2 = 424
424 ÷ 2 = 212
212 ÷ 2 = 106
106 ÷ 2 = 53

Since 53 is a prime number, we have found the prime factors of 848, which are 2 × 2 × 2 × 2 × 53.

Number 2: 130

To find the prime factors of 130, we will start by dividing it by the smallest prime number, which is 2.

130 ÷ 2 = 65
65 ÷ 5 = 13

Since 13 is a prime number, we have found the prime factors of 130, which are 2 × 5 × 13.

Number 3: 462

To find the prime factors of 462, we will start by dividing it by the smallest prime number, which is 2.

462 ÷ 2 = 231
231 ÷ 3 = 77
77 ÷ 7 = 11

Since 11 is a prime number, we have found the prime factors of 462, which are 2 × 3 × 7 × 11.

Step 2: Identify the Common Prime Factors

Now that we have found the prime factors of each of the three numbers, we will identify the common prime factors among them.

The common prime factors among the three numbers are 2.

Step 3: Determine the LCM

To determine the LCM of the three numbers, we will use the common prime factors and the highest power of each prime factor that appears in any of the numbers.

The LCM of 848, 130, and 462 is 2 × 2 × 2 × 5 × 7 × 11 × 13.

Step 4: Determine the HCF

To determine the HCF of the three numbers, we will use the common prime factors and the lowest power of each prime factor that appears in any of the numbers.

The HCF of 848, 130, and 462 is 2.

Conclusion

In this article, we have explored how to determine the LCM and HCF of three given numbers using prime factors. We have found the prime factors of each of the three numbers, identified the common prime factors, and used them to determine the LCM and HCF. The LCM of 848, 130, and 462 is 2 × 2 × 2 × 5 × 7 × 11 × 13, while the HCF is 2.

Example Problems

Problem 1

Find the LCM and HCF of 120, 180, and 240.

Solution

To find the LCM and HCF of 120, 180, and 240, we will follow the same steps as before.

The prime factors of 120 are 2 × 2 × 2 × 3 × 5. The prime factors of 180 are 2 × 2 × 3 × 3 × 5. The prime factors of 240 are 2 × 2 × 2 × 2 × 3 × 5.

The common prime factors among the three numbers are 2 × 2 × 3 × 5.

The LCM of 120, 180, and 240 is 2 × 2 × 2 × 2 × 3 × 3 × 5. The HCF of 120, 180, and 240 is 2 × 2 × 3 × 5.

Problem 2

Find the LCM and HCF of 150, 200, and 300.

Solution

To find the LCM and HCF of 150, 200, and 300, we will follow the same steps as before.

The prime factors of 150 are 2 × 3 × 5 × 5. The prime factors of 200 are 2 × 2 × 2 × 5 × 5. The prime factors of 300 are 2 × 2 × 3 × 5 × 5.

The common prime factors among the three numbers are 2 × 5 × 5.

The LCM of 150, 200, and 300 is 2 × 2 × 2 × 3 × 5 × 5 × 5. The HCF of 150, 200, and 300 is 2 × 5 × 5.

Practice Problems

Problem 1

Find the LCM and HCF of 100, 150, and 200.

Problem 2

Find the LCM and HCF of 120, 180, and 240.

Problem 3

Find the LCM and HCF of 150, 200, and 300.

Answer Key

Problem 1

The LCM of 100, 150, and 200 is 2 × 2 × 5 × 5 × 5. The HCF of 100, 150, and 200 is 2 × 5.

Problem 2

The LCM of 120, 180, and 240 is 2 × 2 × 2 × 3 × 3 × 5. The HCF of 120, 180, and 240 is 2 × 2 × 3 × 5.

Problem 3

Q: What is the Least Common Multiple (LCM)?

A: The Least Common Multiple (LCM) of a set of numbers is the smallest number that is a multiple of each of the numbers in the set.

Q: What is the Highest Common Factor (HCF)?

A: The Highest Common Factor (HCF) of a set of numbers is the largest number that divides each of the numbers in the set without leaving a remainder.

Q: How do I find the LCM and HCF of two numbers?

A: To find the LCM and HCF of two numbers, you can follow these steps:

Find the prime factors of each number: Start by finding the prime factors of each of the two numbers.
Identify the common prime factors: Identify the common prime factors among the two numbers.
Determine the LCM: Use the common prime factors to determine the LCM of the two numbers.
Determine the HCF: Use the common prime factors to determine the HCF of the two numbers.

Q: How do I find the LCM and HCF of three numbers?

A: To find the LCM and HCF of three numbers, you can follow these steps:

Find the prime factors of each number: Start by finding the prime factors of each of the three numbers.
Identify the common prime factors: Identify the common prime factors among the three numbers.
Determine the LCM: Use the common prime factors to determine the LCM of the three numbers.
Determine the HCF: Use the common prime factors to determine the HCF of the three numbers.

Q: What is the difference between LCM and HCF?

A: The main difference between LCM and HCF is that LCM is the smallest number that is a multiple of each of the numbers in the set, while HCF is the largest number that divides each of the numbers in the set without leaving a remainder.

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

A: LCM and HCF are used in various real-life situations, such as:

Music: LCM is used to find the lowest common denominator of two or more notes, while HCF is used to find the highest common factor of two or more notes.
Cooking: LCM is used to find the largest quantity of ingredients that can be used in a recipe, while HCF is used to find the smallest quantity of ingredients that can be used in a recipe.
Science: LCM is used to find the smallest unit of measurement that can be used to measure a quantity, while HCF is used to find the largest unit of measurement that can be used to measure a quantity.

Q: Can I use LCM and HCF with negative numbers?

A: Yes, you can use LCM and HCF with negative numbers. However, you need to take into account the fact that negative numbers have a different sign than positive numbers.

Q: Can I use LCM and HCF with fractions?

A: Yes, you can use LCM and HCF with fractions. However, you need to take into account the fact that fractions have a different form than whole numbers.

Q: How do I find the LCM and HCF of a set of numbers with different bases?

A: To find the LCM and HCF of a set of numbers with different bases, you need to convert each number to a common base, such as base 10. Then, you can follow the same steps as before to find the LCM and HCF.

Q: Can I use LCM and HCF with complex numbers?

A: Yes, you can use LCM and HCF with complex numbers. However, you need to take into account the fact that complex numbers have a different form than real numbers.

Q: How do I find the LCM and HCF of a set of numbers with different exponents?

A: To find the LCM and HCF of a set of numbers with different exponents, you need to take into account the fact that the exponents are different. Then, you can follow the same steps as before to find the LCM and HCF.

Q: Can I use LCM and HCF with irrational numbers?

A: Yes, you can use LCM and HCF with irrational numbers. However, you need to take into account the fact that irrational numbers have a different form than rational numbers.

Q: How do I find the LCM and HCF of a set of numbers with different units?

A: To find the LCM and HCF of a set of numbers with different units, you need to convert each number to a common unit, such as meters. Then, you can follow the same steps as before to find the LCM and HCF.

Q: Can I use LCM and HCF with mixed numbers?

A: Yes, you can use LCM and HCF with mixed numbers. However, you need to take into account the fact that mixed numbers have a different form than whole numbers.

Q: How do I find the LCM and HCF of a set of numbers with different scales?

A: To find the LCM and HCF of a set of numbers with different scales, you need to convert each number to a common scale, such as a logarithmic scale. Then, you can follow the same steps as before to find the LCM and HCF.

Q: Can I use LCM and HCF with decimals?

A: Yes, you can use LCM and HCF with decimals. However, you need to take into account the fact that decimals have a different form than whole numbers.

Q: How do I find the LCM and HCF of a set of numbers with different precision?

A: To find the LCM and HCF of a set of numbers with different precision, you need to take into account the fact that the precision is different. Then, you can follow the same steps as before to find the LCM and HCF.

Q: Can I use LCM and HCF with percentages?

A: Yes, you can use LCM and HCF with percentages. However, you need to take into account the fact that percentages have a different form than whole numbers.

Q: How do I find the LCM and HCF of a set of numbers with different units of measurement?

A: To find the LCM and HCF of a set of numbers with different units of measurement, you need to convert each number to a common unit of measurement, such as meters. Then, you can follow the same steps as before to find the LCM and HCF.

Q: Can I use LCM and HCF with time?

A: Yes, you can use LCM and HCF with time. However, you need to take into account the fact that time has a different form than other units of measurement.

Q: How do I find the LCM and HCF of a set of numbers with different time units?

A: To find the LCM and HCF of a set of numbers with different time units, you need to convert each number to a common time unit, such as seconds. Then, you can follow the same steps as before to find the LCM and HCF.

Q: Can I use LCM and HCF with money?

A: Yes, you can use LCM and HCF with money. However, you need to take into account the fact that money has a different form than other units of measurement.

Q: How do I find the LCM and HCF of a set of numbers with different currency units?

A: To find the LCM and HCF of a set of numbers with different currency units, you need to convert each number to a common currency unit, such as dollars. Then, you can follow the same steps as before to find the LCM and HCF.

Q: Can I use LCM and HCF with temperature?

A: Yes, you can use LCM and HCF with temperature. However, you need to take into account the fact that temperature has a different form than other units of measurement.

Q: How do I find the LCM and HCF of a set of numbers with different temperature units?

A: To find the LCM and HCF of a set of numbers with different temperature units, you need to convert each number to a common temperature unit, such as Celsius. Then, you can follow the same steps as before to find the LCM and HCF.

Q: Can I use LCM and HCF with volume?

A: Yes, you can use LCM and HCF with volume. However, you need to take into account the fact that volume has a different form than other units of measurement.

Q: How do I find the LCM and HCF of a set of numbers with different volume units?

A: To find the LCM and HCF of a set of numbers with different volume units, you need to convert each number to a common volume unit, such as liters. Then, you can follow the same steps as before to find the LCM and HCF.

Q: Can I use LCM and HCF with area?

A: Yes