Given:$\[ A = 2 \times 3^4 \times 5 \\]$\[ B = 2^3 \times 3^2 \times 5^2 \\]b) Write Down The Highest Common Factor (HCF) Of \[$ A \$\] And \[$ B \$\].$\[ HCF = \square \\]

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In mathematics, the highest common factor (HCF) of two numbers is the largest positive integer that divides both numbers without leaving a remainder. In this article, we will discuss how to find the HCF of two given numbers, A and B.

Understanding the Given Numbers

We are given two numbers:

  • A = 2 × 3^4 × 5
  • B = 2^3 × 3^2 × 5^2

To find the HCF of A and B, we need to first understand the prime factorization of both numbers.

Prime Factorization of A and B

The prime factorization of A is:

2 × 3^4 × 5

The prime factorization of B is:

2^3 × 3^2 × 5^2

Finding the HCF

To find the HCF of A and B, we need to find the common prime factors between the two numbers and take the lowest power of each common factor.

The common prime factors between A and B are:

  • 2
  • 3
  • 5

The lowest power of 2 in A is 2^1 (since 2 is raised to the power of 1 in A), and the lowest power of 2 in B is 2^3. Therefore, the lowest power of 2 that is common to both A and B is 2^1.

The lowest power of 3 in A is 3^4, and the lowest power of 3 in B is 3^2. Therefore, the lowest power of 3 that is common to both A and B is 3^2.

The lowest power of 5 in A is 5^1, and the lowest power of 5 in B is 5^2. Therefore, the lowest power of 5 that is common to both A and B is 5^1.

Calculating the HCF

Now that we have found the common prime factors and their lowest powers, we can calculate the HCF by multiplying these factors together:

HCF = 2^1 × 3^2 × 5^1 = 2 × 9 × 5 = 90

Therefore, the highest common factor (HCF) of A and B is 90.

Conclusion

In this article, we discussed how to find the highest common factor (HCF) of two numbers, A and B. We first understood the prime factorization of both numbers, then found the common prime factors and their lowest powers, and finally calculated the HCF by multiplying these factors together. The HCF of A and B is 90.

Example Use Cases

The HCF of two numbers can be used in various real-world applications, such as:

  • Simplifying fractions: The HCF of two numbers can be used to simplify fractions by dividing both the numerator and the denominator by their HCF.
  • Finding the greatest common divisor (GCD): The HCF of two numbers is equal to their greatest common divisor (GCD).
  • Solving equations: The HCF of two numbers can be used to solve equations by finding the common factors between the two numbers.

Common Mistakes to Avoid

When finding the HCF of two numbers, there are several common mistakes to avoid:

  • Not understanding the prime factorization: It is essential to understand the prime factorization of both numbers to find their HCF.
  • Not finding the lowest power of each common factor: The HCF is calculated by taking the lowest power of each common factor.
  • Not multiplying the common factors together: The HCF is calculated by multiplying the common factors together.

In this article, we will answer some frequently asked questions about finding the highest common factor (HCF) of two numbers.

Q: What is the highest common factor (HCF) of two numbers?

A: The highest common factor (HCF) of two numbers is the largest positive integer that divides both numbers without leaving a remainder.

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

A: To find the HCF of two numbers, you need to follow these steps:

  1. Understand the prime factorization: Understand the prime factorization of both numbers.
  2. Find the common prime factors: Find the common prime factors between the two numbers.
  3. Take the lowest power of each common factor: Take the lowest power of each common factor.
  4. Multiply the common factors together: Multiply the common factors together to find the HCF.

Q: What is the difference between the HCF and the least common multiple (LCM)?

A: The HCF and the least common multiple (LCM) are two related but distinct concepts.

  • HCF: The HCF of two numbers is the largest positive integer that divides both numbers without leaving a remainder.
  • LCM: The LCM of two numbers is the smallest positive integer that is a multiple of both numbers.

Q: How do I use the HCF in real-world applications?

A: The HCF of two numbers can be used in various real-world applications, such as:

  • Simplifying fractions: The HCF of two numbers can be used to simplify fractions by dividing both the numerator and the denominator by their HCF.
  • Finding the greatest common divisor (GCD): The HCF of two numbers is equal to their greatest common divisor (GCD).
  • Solving equations: The HCF of two numbers can be used to solve equations by finding the common factors between the two numbers.

Q: What are some common mistakes to avoid when finding the HCF?

A: When finding the HCF of two numbers, there are several common mistakes to avoid:

  • Not understanding the prime factorization: It is essential to understand the prime factorization of both numbers to find their HCF.
  • Not finding the lowest power of each common factor: The HCF is calculated by taking the lowest power of each common factor.
  • Not multiplying the common factors together: The HCF is calculated by multiplying the common factors together.

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

A: Yes, you can use a calculator to find the HCF of two numbers. However, it is essential to understand the prime factorization of both numbers and how to calculate the HCF manually.

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

A: To find the HCF of three or more numbers, you can follow these steps:

  1. Find the HCF of two numbers: Find the HCF of two numbers.
  2. Find the HCF of the result and the third number: Find the HCF of the result and the third number.
  3. Continue this process: Continue this process until you have found the HCF of all the numbers.

Q: Can I use the HCF to find the LCM?

A: Yes, you can use the HCF to find the LCM. The LCM of two numbers is equal to the product of the two numbers divided by their HCF.

Conclusion

In this article, we have answered some frequently asked questions about finding the highest common factor (HCF) of two numbers. We have discussed the definition of the HCF, how to find it, and its applications in real-world scenarios. We have also highlighted some common mistakes to avoid when finding the HCF and provided tips on how to use the HCF to find the least common multiple (LCM).