If The HCF Of Two Positive Integers A And B Is 1, Then Their LCM Is: (A) a+b (B) a (C) b (D) ab
Understanding the Relationship Between HCF and LCM
The relationship between the Highest Common Factor (HCF) and the Least Common Multiple (LCM) of two positive integers is a fundamental concept in mathematics. The HCF of two numbers is the greatest number that divides both of them without leaving a remainder, while the LCM is the smallest number that is a multiple of both numbers. In this article, we will explore the relationship between HCF and LCM, and specifically, we will determine the LCM of two positive integers when their HCF is 1.
Theorem: If the HCF of two positive integers a and b is 1, then their LCM is ab
To understand this theorem, let's first recall the definition of HCF and LCM. The HCF of two numbers a and b is the greatest number that divides both a and b without leaving a remainder. On the other hand, the LCM of a and b is the smallest number that is a multiple of both a and b.
When the HCF of two numbers is 1, it means that the only common factor between them is 1. In other words, they have no other common factors other than 1. This implies that the two numbers are relatively prime, meaning they have no common factors other than 1.
Proof of the Theorem
To prove the theorem, we can use the following steps:
- Assume that the HCF of two positive integers a and b is 1.
- Since the HCF of a and b is 1, we know that a and b are relatively prime.
- By definition, the LCM of a and b is the smallest number that is a multiple of both a and b.
- Since a and b are relatively prime, the only common factor between them is 1.
- Therefore, the smallest number that is a multiple of both a and b is their product, ab.
- Hence, the LCM of a and b is ab.
Example
Let's consider an example to illustrate this theorem. Suppose we have two positive integers, a = 6 and b = 15. The HCF of 6 and 15 is 1, since they have no common factors other than 1.
To find the LCM of 6 and 15, we can use the theorem. Since the HCF of 6 and 15 is 1, we know that their LCM is ab = 6 × 15 = 90.
Conclusion
In conclusion, if the HCF of two positive integers a and b is 1, then their LCM is ab. This theorem is a fundamental concept in mathematics and has numerous applications in various fields, including number theory, algebra, and combinatorics.
Frequently Asked Questions
- What is the relationship between HCF and LCM?
- If the HCF of two positive integers a and b is 1, what is their LCM?
- How do we find the LCM of two numbers when their HCF is 1?
Answer
- The relationship between HCF and LCM is that the HCF of two numbers is the greatest number that divides both of them without leaving a remainder, while the LCM is the smallest number that is a multiple of both numbers.
- If the HCF of two positive integers a and b is 1, then their LCM is ab.
- To find the LCM of two numbers when their HCF is 1, we can use the theorem that states that the LCM of two relatively prime numbers is their product.
References
- [1] "Number Theory" by G.H. Hardy and E.M. Wright
- [2] "Algebra" by Michael Artin
- [3] "Combinatorics" by Richard P. Stanley
Further Reading
- "The HCF and LCM of Two Numbers"
- "The Relationship Between HCF and LCM"
- "The LCM of Two Relatively Prime Numbers"
Final Thoughts
In conclusion, the relationship between HCF and LCM is a fundamental concept in mathematics. The theorem that states that if the HCF of two positive integers a and b is 1, then their LCM is ab, is a powerful tool that has numerous applications in various fields. By understanding this theorem, we can find the LCM of two numbers when their HCF is 1, and apply it to solve various mathematical problems.
Q&A: Frequently Asked Questions and Answers
In this section, we will address some of the most frequently asked questions related to the relationship between HCF and LCM, and specifically, the theorem that states that if the HCF of two positive integers a and b is 1, then their LCM is ab.
Q: What is the relationship between HCF and LCM?
A: The relationship between HCF and LCM is that the HCF of two numbers is the greatest number that divides both of them without leaving a remainder, while the LCM is the smallest number that is a multiple of both numbers.
Q: If the HCF of two positive integers a and b is 1, what is their LCM?
A: If the HCF of two positive integers a and b is 1, then their LCM is ab.
Q: How do we find the LCM of two numbers when their HCF is 1?
A: To find the LCM of two numbers when their HCF is 1, we can use the theorem that states that the LCM of two relatively prime numbers is their product.
Q: What is the significance of the HCF being 1 in the context of LCM?
A: The significance of the HCF being 1 is that it implies that the two numbers are relatively prime, meaning they have no common factors other than 1. This makes it easier to find the LCM of the two numbers.
Q: Can we find the LCM of two numbers if their HCF is not 1?
A: Yes, we can find the LCM of two numbers if their HCF is not 1. However, it may be more challenging to find the LCM in this case.
Q: How do we apply the theorem in real-world scenarios?
A: The theorem can be applied in various real-world scenarios, such as finding the LCM of two numbers in physics, engineering, or computer science.
Q: What are some common mistakes to avoid when finding the LCM of two numbers?
A: Some common mistakes to avoid when finding the LCM of two numbers include:
- Not considering the HCF of the two numbers
- Not using the correct formula for finding the LCM
- Not checking for common factors between the two numbers
Q: Can we find the LCM of three or more numbers?
A: Yes, we can find the LCM of three or more numbers. However, it may be more challenging to find the LCM in this case.
Q: How do we find the LCM of three or more numbers?
A: To find the LCM of three or more numbers, we can use the following steps:
- Find the LCM of the first two numbers
- Find the LCM of the result and the third number
- Continue this process until we have found the LCM of all the numbers
Q: What are some real-world applications of the theorem?
A: Some real-world applications of the theorem include:
- Finding the LCM of two numbers in physics, engineering, or computer science
- Solving problems involving the LCM of two numbers in mathematics
- Applying the theorem in cryptography and coding theory
Conclusion
In conclusion, the relationship between HCF and LCM is a fundamental concept in mathematics. The theorem that states that if the HCF of two positive integers a and b is 1, then their LCM is ab, is a powerful tool that has numerous applications in various fields. By understanding this theorem, we can find the LCM of two numbers when their HCF is 1, and apply it to solve various mathematical problems.
Frequently Asked Questions
- What is the relationship between HCF and LCM?
- If the HCF of two positive integers a and b is 1, what is their LCM?
- How do we find the LCM of two numbers when their HCF is 1?
Answer
- The relationship between HCF and LCM is that the HCF of two numbers is the greatest number that divides both of them without leaving a remainder, while the LCM is the smallest number that is a multiple of both numbers.
- If the HCF of two positive integers a and b is 1, then their LCM is ab.
- To find the LCM of two numbers when their HCF is 1, we can use the theorem that states that the LCM of two relatively prime numbers is their product.
References
- [1] "Number Theory" by G.H. Hardy and E.M. Wright
- [2] "Algebra" by Michael Artin
- [3] "Combinatorics" by Richard P. Stanley
Further Reading
- "The HCF and LCM of Two Numbers"
- "The Relationship Between HCF and LCM"
- "The LCM of Two Relatively Prime Numbers"
Final Thoughts
In conclusion, the relationship between HCF and LCM is a fundamental concept in mathematics. The theorem that states that if the HCF of two positive integers a and b is 1, then their LCM is ab, is a powerful tool that has numerous applications in various fields. By understanding this theorem, we can find the LCM of two numbers when their HCF is 1, and apply it to solve various mathematical problems.