Numbers with Specific HCF and LCM: Understanding the Relationship Between Highest Common Factor and Least Common Multiple
Understanding the relationship between the highest common factor (HCF) and the least common multiple (LCM) is crucial in number theory, especially when dealing with complex mathematical problems. In this article, we aim to explore a specific scenario where we need to find two numbers that have an HCF of 20 and an LCM of 140. We will use the formula HCF × LCM Product of the two numbers to solve this problem.
Introduction to HCF and LCM
The highest common factor (HCF) of two or more numbers is the largest number that divides each of them without leaving a remainder. On the other hand, the least common multiple (LCM) of two or more integers is the smallest positive integer that is divisible by each of the numbers.
Using the Formula to Find the Numbers
Given that we need to find two numbers with an HCF of 20 and an LCM of 140, we can use the formula:
HCF × LCM Product of the two numbers
Substituting the given values:
20 × 140 Product of the two numbers
Product 2800
Now, we need to find two numbers whose product is 2800 and whose HCF is 20.
Factorization and Verification
Let's factorize 2800:
2800 20 × 140
Therefore, the two numbers are 40 and 70, as:
40 × 70 2800
And the HCF of 40 and 70 is 20.
Common Mistakes and Clarifications
It is important to note that this problem has been constructed with specific constraints to ensure mathematical consistency. Here are some clarifications and common mistakes to avoid:
Mistake 1: Incorrect LCM and HCF
The user initial proposed numbers 280 and 140 and another set 14 and 140, which are incorrect because 14 cannot be the LCM of 280 and 140, and the LCM of 14 and 140 should be 140 itself, not 20.
Mistake 2: Prime Factorization Error
The user also suggested that the only numbers with an LCM of 14 are 2 and 7, which is incorrect. These numbers are indeed prime, but their LCM is 14, and their HCF is 1, not 20. Additionally, the LCM is always a multiple of the HCF, and 140 should be a multiple of 20, which is true.
Mistake 3: Reverse HCF and LCM Relationship
The user suggested that the HCF cannot be larger than the LCM. This is true, as the HCF is always less than or equal to the LCM. However, the user should have used the correct numbers that fit the condition of having an HCF of 20 and an LCM of 140.
Conclusion
In summary, the two numbers with an HCF of 20 and an LCM of 140 are 40 and 70. This example highlights the importance of understanding number theory concepts and the relationships between HCF and LCM. Always ensure that the given conditions are met for the problems to be mathematically valid and consistent.