Understanding the HCF and LCM of Numbers

This article will explore how to find two numbers that have a specific Highest Common Factor (HCF) and Lowest Common Multiple (LCM). We will cover the relationship between HCF and LCM and provide a step-by-step method to solve such problems. The specific example we will explore is finding two numbers whose HCF is 6 and LCM is 60.

The Relationship Between HCF and LCM

The relationship between the HCF and LCM of two numbers is given by the formula:

[ HCF(a, b) times LCM(a, b) a times b ]

This relationship is crucial in solving the problem at hand. In this context, we need to find two numbers (a) and (b) such that:

HCF((a), (b)) 6 LCM((a), (b)) 60

Step-by-Step Solution

Let's assume the two numbers are (a) and (b). We start by using the relationship between HCF and LCM:

[ 6 times 60 a times b ]

This simplifies to:

[ 360 a times b ]

Since the HCF of the numbers is 6, we can express (a) and (b) in terms of their HCF (6). Let:

[ a 6m text{ and } b 6n ]

where (m) and (n) are coprime (i.e., their HCF is 1).

Substituting these into the equation (a times b 360), we get:

[ (6m) times (6n) 360 ]

This simplifies to:

[ 36mn 360 ]

Dividing both sides by 36, we have:

[ mn 10 ]

Now we need to find pairs of coprime numbers (m) and (n) such that their product is 10.

The possible pairs are:

1 and 10 2 and 5

Let's calculate the corresponding numbers (a) and (b) for each pair:

Pair 1: 1 and 10

For (m 1) and (n 10): (a 6 times 1 6) (b 6 times 10 60)

Pair 2: 2 and 5

For (m 2) and (n 5): (a 6 times 2 12) (b 6 times 5 30)

Thus, the pairs of numbers that satisfy the conditions are:

6 and 60 12 and 30

To verify, let's check the HCF and LCM of these pairs:

HCF(6, 60) 6 and LCM(6, 60) 60 HCF(12, 30) 6 and LCM(12, 30) 60

Both pairs meet the criteria, confirming their validity.

Additional Examples and Explanation

Starting with the known values:

[ LCM div HCF 60 div 6 10 ]

The factors of 10 are 1 and 10, and 2 and 5. The pairs of coprime factors are 2 and 5. Multiplying these by the HCF (6) gives us the pairs of numbers:

1st number 2 (times 6 12) 2nd number 5 (times 6 30)

Also, another valid pair is:

1st number 6 2nd number 60

Both sets of numbers meet the criteria of having an HCF of 6 and an LCM of 60.

Conclusion

By using the relationship between HCF and LCM, we can solve for the numbers that satisfy the given conditions. This method not only helps in understanding the underlying mathematical principles but also provides practical steps to solve similar problems.