Understanding LCM and HCF in Number Theory Problems: A Practical Example
In number theory, the Least Common Multiple (LCM) and the Highest Common Factor (HCF) are important concepts that help solve various problems. This article will delve into a specific problem that involves both LCM and HCF, providing detailed explanations and a step-by-step approach to solving it.
The Problem and Initial Analysis
Given the LCM (68) and the HCF (5) of two numbers, and one of the numbers (20), the task is to find the other number. Initially, one may attempt to solve it using the equation:
Initial Approach
685/2017
Analysis of the Initial Approach
The analysis shows that 17 is the number obtained, and since the LCM is 68 and the HCF is 5, it might seem that 17 is the correct answer.
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17
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However, there is a critical consideration: 20 and 17 cannot have 5 as the HCF. Furthermore, 20 and 17 cannot have 68 as the LCM. This indicates that the initial solution is problematic. 17 is a prime number and cannot have factors other than 1 and itself.
Correct Approach
Given the LCM (68) and the HCF (5), and knowing one of the numbers (20), we can use the relationship between LCM, HCF, and the two numbers. The relationship is:
LCM x HCF Product of the numbers.
Substituting the given values:
68 x 5 340
This means the product of the two numbers is 340. Given that one number is 20, the other number can be found by dividing the product by 20:
Correct Calculation
The second number 68 x 5 / 20
68 / 4
17
To confirm, let's check the initial equation again:
5681720
Other number 5 x 68 / 20
340 / 20
17
Explanation of the Solution
The correct solution uses the relationship that the LCM of two numbers multiplied by their HCF equals the product of the two numbers. Given the LCM is 68 and the HCF is 5, the product of the numbers is 340. Dividing this product by the known number (20) gives the other number, which is 17.
Conclusion
This example illustrates the importance of verifying the consistency of the HCF and LCM with the given numbers. It also demonstrates the application of the relationship between LCM, HCF, and the product of the numbers in solving such problems.
Related Queries
Q: What is the least common multiple (LCM)?
LCM is the smallest number that is a multiple of two or more numbers.
Q: What is the highest common factor (HCF)?
HCF is the largest number that divides two or more numbers without leaving a remainder.
Q: How do you find the LCM and HCF of two numbers?
To find the LCM, you can use the prime factorization method or the division method. To find the HCF, you can use the prime factorization method, the Euclidean algorithm, or the division method.