Understanding the Relationship Between HCF and LCM

In this article, we will explore how to find two numbers given their Highest Common Factor (HCF) and Lowest Common Multiple (LCM). We will use the relationship between these factors to discover the numbers. This article is targeted for SEO optimization by Google, making it easily discoverable and useful for students, teachers, and anyone seeking to enhance their problem-solving skills.

The Formula in Use

We start with the fundamental formula that relates the HCF and LCM to the product of two numbers:

HCF ? LCM a ? b

Given Conditions

The problem at hand is to find two numbers a and b such that:

HCF(a, b) 20 LCM(a, b) 300

Setting Up the Equation

Given the HCF and LCM, we can set up the equation:

20 ? 300 a ? b

This simplifies to:

6000 a ? b

Expressing a and b in Terms of Their HCF

A number can be expressed as a product of HCF and another integer. Therefore, we can write:

a 20m

b 20n

where m and n are coprime integers (their HCF is 1).

Solving for m and n

Substituting these expressions into the equation for the product, we get:

20m ? 20n 6000

This simplifies to:

400mn 6000

Dividing both sides by 400, we get:

mn 15

Finding Coprime Integers m and n

Now, we need to find pairs of coprime integers m and n such that their product is 15. The pairs of factors of 15 are:

1 ? 15 3 ? 5

Calculating a and b for Each Pair

We can calculate a and b for each pair:

For m 1 and n 15: a 20 ? 1 20 b 20 ? 15 300 For m 3 and n 5: a 20 ? 3 60 b 20 ? 5 100

Verification

Both pairs of numbers (20, 300) and (60, 100) satisfy the conditions:

HCF 20 LCM 300

Prime Factorization Verification

We can verify the prime factorization for the given HCF and LCM:

20 2 ? 2 ? 5

300 2 ? 2 ? 5 ? 5 ? 3

100 2 ? 2 ? 5 ? 5

60 2 ? 2 ? 3 ? 5

The LCM is correctly derived by taking the highest power of each prime factor present in the numbers:

LCM 2 ? 2 ? 5 ? 5 ? 3 300

The HCF is correctly derived by taking the common prime factors with the lowest power:

HCF 2 ? 2 ? 5 20

Conclusion

In conclusion, the two pairs of numbers that satisfy the given conditions are:

20 and 300 60 and 100

These pairs demonstrate the application of the relationship between HCF and LCM, providing a systematic approach to solving number problems.