Highest Common Factor Of 6a3bc2, 21a²b And 15a³ Is (a) 3a² \(b) 3a³ (c) 6a³ (d) 6a
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Introduction
The Highest Common Factor (HCF) of a set of numbers is the largest positive integer that divides each of the numbers in the set without leaving a remainder. In this article, we will find the HCF of three given numbers: 6a3b2, 21a^2b, and 15a^3.
Step 1: Factorize the Numbers
To find the HCF, we need to factorize each number into its prime factors.
- 6a3b2 = 2 × 3 × a^3 × b^2
- 21a^2b = 3 × 7 × a^2 × b
- 15a^3 = 3 × 5 × a^3
Step 2: Identify Common Factors
Now, we need to identify the common factors among the three numbers.
- The common factors are 3 and a^2 (since a^2 is a factor of a^3).
Step 3: Determine the HCF
To determine the HCF, we need to multiply the common factors.
- HCF = 3 × a^2 = 3a^2
Conclusion
Therefore, the HCF of 6a3b2, 21a^2b, and 15a^3 is 3a^2.
Frequently Asked Questions (FAQs)
Q: What is the Highest Common Factor (HCF)?
A: The HCF of a set of numbers is the largest positive integer that divides each of the numbers in the set without leaving a remainder.
Q: How do we find the HCF of a set of numbers?
A: To find the HCF, we need to factorize each number into its prime factors and then identify the common factors. Finally, we multiply the common factors to determine the HCF.
Q: What is the HCF of 6a3b2, 21a^2b, and 15a^3?
A: The HCF of 6a3b2, 21a^2b, and 15a^3 is 3a^2.
Key Takeaways
- The HCF of a set of numbers is the largest positive integer that divides each of the numbers in the set without leaving a remainder.
- To find the HCF, we need to factorize each number into its prime factors and then identify the common factors.
- The HCF of 6a3b2, 21a^2b, and 15a^3 is 3a^2.
Practice Problems
Find the HCF of the following sets of numbers:
- 12a^2b, 18a3b2, and 24a^4b
- 15a2b2, 20a^3b, and 25a4b2
Solutions
Solution 1
- 12a^2b = 2^2 × 3 × a^2 × b
- 18a3b2 = 2 × 3^2 × a^3 × b^2
- 24a^4b = 2^3 × 3 × a^4 × b
- Common factors: 2, 3, and a^2
- HCF = 2 × 3 × a^2 = 6a^2
Solution 2
- 15a2b2 = 3 × 5 × a^2 × b^2
- 20a^3b = 2^2 × 5 × a^3 × b
- 25a4b2 = 5^2 × a^4 × b^2
- Common factors: 5 and a^2
- HCF = 5 × a^2 = 5a^2
Conclusion
In this article, we learned how to find the Highest Common Factor (HCF) of a set of numbers. We also solved two practice problems to reinforce our understanding of the concept. Remember, the HCF of a set of numbers is the largest positive integer that divides each of the numbers in the set without leaving a remainder.
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Frequently Asked Questions (FAQs)
Q: What is the Highest Common Factor (HCF)?
A: The HCF of a set of numbers is the largest positive integer that divides each of the numbers in the set without leaving a remainder.
Q: How do we find the HCF of a set of numbers?
A: To find the HCF, we need to factorize each number into its prime factors and then identify the common factors. Finally, we multiply the common factors to determine the HCF.
Q: What is the difference between HCF and LCM?
A: The HCF of a set of numbers is the largest positive integer that divides each of the numbers in the set without leaving a remainder, while the Least Common Multiple (LCM) is the smallest positive integer that is a multiple of each of the numbers in the set.
Q: How do we find the LCM of a set of numbers?
A: To find the LCM, we need to find the product of the highest powers of all the prime factors involved in the numbers.
Q: What is the relationship between HCF and LCM?
A: The product of the HCF and LCM of two numbers is equal to the product of the two numbers.
Q: Can the HCF of a set of numbers be zero?
A: No, the HCF of a set of numbers cannot be zero.
Q: Can the HCF of a set of numbers be negative?
A: No, the HCF of a set of numbers cannot be negative.
Q: How do we find the HCF of a set of numbers with variables?
A: To find the HCF of a set of numbers with variables, we need to factorize each number into its prime factors and then identify the common factors. Finally, we multiply the common factors to determine the HCF.
Q: What is the HCF of 6a3b2, 21a^2b, and 15a^3?
A: The HCF of 6a3b2, 21a^2b, and 15a^3 is 3a^2.
Q: What is the HCF of 12a^2b, 18a3b2, and 24a^4b?
A: The HCF of 12a^2b, 18a3b2, and 24a^4b is 6a^2b.
Q: What is the HCF of 15a2b2, 20a^3b, and 25a4b2?
A: The HCF of 15a2b2, 20a^3b, and 25a4b2 is 5a^2b.
Key Takeaways
- The HCF of a set of numbers is the largest positive integer that divides each of the numbers in the set without leaving a remainder.
- To find the HCF, we need to factorize each number into its prime factors and then identify the common factors.
- The HCF of a set of numbers with variables is found by factorizing each number into its prime factors and then identifying the common factors.
Practice Problems
Find the HCF of the following sets of numbers:
- 24a3b2, 36a4b3, and 48a5b4
- 18a2b2, 24a3b3, and 30a4b4
Solutions
Solution 1
- 24a3b2 = 2^3 × 3 × a^3 × b^2
- 36a4b3 = 2^2 × 3^2 × a^4 × b^3
- 48a5b4 = 2^4 × 3 × a^5 × b^4
- Common factors: 2, 3, and a^3
- HCF = 2 × 3 × a^3 = 6a^3
Solution 2
- 18a2b2 = 2 × 3^2 × a^2 × b^2
- 24a3b3 = 2^3 × 3 × a^3 × b^3
- 30a4b4 = 2 × 3 × 5 × a^4 × b^4
- Common factors: 2, 3, and a^2
- HCF = 2 × 3 × a^2 = 6a^2
Conclusion
In this article, we answered some frequently asked questions about the Highest Common Factor (HCF). We also solved two practice problems to reinforce our understanding of the concept. Remember, the HCF of a set of numbers is the largest positive integer that divides each of the numbers in the set without leaving a remainder.