How to Find Two Numbers When Their HCF is 6 and LCM is 60

The problem of finding two numbers given their Highest Common Factor (HCF) and Least Common Multiple (LCM) is a common question in mathematics. This article will guide you through the process of finding two numbers whose HCF is 6 and LCM is 60. By understanding the fundamental relationship between HCF and LCM, you can systematically solve a variety of similar problems.

Understanding the Relationship between HCF and LCM

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

HCF(a, b) × LCM(a, b) a × b

This relationship can be used to find the two numbers when you know their HCF and LCM. Let's break down the steps to solve the problem:

Step-by-Step Process

Step 1: Calculate the Product of the Two Numbers

Given:

HCF 6

LCM 60

Using the formula:

HCF × LCM a × b

We obtain:

6 × 60 a × b

This simplifies to:

360 a × b

Step 2: Express the Numbers in Terms of Their HCF

Since the HCF is 6, we can express the two numbers as:

a 6m and b 6n

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

Step 3: Substitute into the Product Equation

Substituting a and b into the product equation:

6m × 6n 360

This simplifies to:

36mn 360

Dividing both sides by 36 gives:

mn 10

Step 4: Find Pairs of Coprime Integers

Now we need to find pairs of integers m and n such that their product is 10 and they are coprime:

m 1 and n 10 m 2 and n 5

Step 5: Calculate the Corresponding Numbers

For each pair, calculate a and b:

For m 1 and n 10: a 6 × 1 6 b 6 × 10 60 For m 2 and n 5: a 6 × 2 12 b 6 × 5 30

Conclusion

The pairs of numbers that satisfy the given conditions (HCF of 6 and LCM of 60) are:

6 and 60 12 and 30

Additional Considerations

When the larger number is a multiple of the smaller number, the HCF is the smaller number and the LCM is the larger number. Therefore, the valid pairs are 12 and 30, and 6 and 60. Other combinations like 24 and 15 are not valid because they don't satisfy the HCF and LCM conditions. Similarly, 3120 cannot be considered because 120 is not a factor of 60.

Additional Examples

Let's consider a few more examples to solidify your understanding:

Example 1: Given HCF 6 and LCM 60, the possible pairs are 6 and 60, 12 and 30.

Example 2: Given HCF 6 and LCM 120, we find factors of 120 ÷ 6 20, and the coprime pairs are (1, 20) and (2, 10). Thus, the valid pairs are (6, 120) and (12, 60).

By following these steps and understanding the relationship between HCF and LCM, you can solve a wide range of similar problems in mathematics.