Understanding the Relationship Between HCF and LCM Through Examples

In mathematics, understanding the relationship between the Highest Common Factor (HCF) and the Least Common Multiple (LCM) is crucial, especially in number theory and problem-solving. This article covers how to find the other number when given the HCF and LCM along with one of the numbers. We'll explore several examples and provide detailed solutions using both algebraic and logical methods.

Example 1: Finding the Other Number Given HCF and LCM

Problem: The HCF of two numbers is 40 and their LCM is 1200. If one of the numbers is 200, what is the other number?

Solution:

We know that HCF * LCM N1 * N2. Here, HCF 40, LCM 1200, and N1 200. To find N2, we can use the formula: (N2 frac{HCF * LCM}{N1}). (N2 frac{40 * 1200}{200} 240).

The second number is 240.

Example 2: Detailed Explanation

Let's break down the logic using an algebraic approach.

Given:

HCF 42 LCM 1260 One number 210

We use the relationship: HCF * LCM N1 * N2.

Substitute the values:

42 * 1260 210 * N2

Solving for N2:

N2 (frac{42 * 1260}{210})

N2 42 * 6 252.

Therefore, the other number is 252.

Example 3: Another Method

Again, using the relation HCF * LCM N1 * N2:

Given:

HCF 42 LCM 1260 One number 210

N2 (frac{HCF * LCM}{N1})

N2 (frac{42 * 1260}{210})

N2 42 * 6 252.

The other number is 252.

Example 4: Comprehensive Solution

Given:

HCF 42 LCM 1260 One number 210

We know that:

First number (A) * Second number (B) HCF * LCM

210 * B 42 * 1260

Solving for B:

B (frac{42 * 1260}{210})

B 252.

Therefore, the other number is 252.

Example 5: Addressing a Misconception

Problem: The HCF of two numbers is 42 and their LCM is 4641. If one of the numbers is between 200 and 300, what is the other number?

Solution:

The given values are HCF 42 and LCM 4641.

Let the number between 200 and 300 be (x).

Using the relationship: HCF * LCM N1 * N2

42 * 4641 x * N2

We need to find (x).

(x 42 * 4641 / N2)

For (x) to be between 200 and 300, it must satisfy this condition.

Upon closer inspection, there seems to be a contradiction, as HCF must be even, but LCM is odd, which is generally not possible. Further investigation may be needed to resolve this issue.

Conclusion

Understanding the relationship between HCF and LCM is essential for solving various mathematical problems. In these examples, we explored how to find the other number given HCF, LCM, and one of the numbers. The methods used include direct calculation and algebraic manipulation. For any unresolved questions, feel free to comment or reach out for clarification.