What is the Relationship between HCF, LCM and Product of Numbers?

S3-SA4-0455

Grade Level:

Class 6

AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering

Definition

What is it?

The relationship between HCF (Highest Common Factor), LCM (Lowest Common Multiple), and the product of two numbers is a fundamental rule in mathematics. It states that for any two positive numbers, the product of their HCF and LCM is always equal to the product of the numbers themselves. This means HCF(a, b) × LCM(a, b) = a × b.

Simple Example

Quick Example

Imagine you have two friends, Rohan and Priya, and their favourite cricket jersey numbers are 6 and 10. If you multiply their jersey numbers (6 × 10 = 60), this product will be the same as multiplying the HCF of 6 and 10 by the LCM of 6 and 10. Let's check: HCF(6,10) is 2, and LCM(6,10) is 30. And 2 × 30 = 60! It works!

Worked Example

Step-by-Step

Let's find the HCF and LCM of 12 and 18, and then verify the relationship.

1. Find the prime factors of 12: 12 = 2 × 2 × 3
---
2. Find the prime factors of 18: 18 = 2 × 3 × 3
---
3. To find HCF: Take the common prime factors with the lowest power. Common factors are 2 and 3. So, HCF(12, 18) = 2 × 3 = 6
---
4. To find LCM: Take all prime factors (common and uncommon) with their highest power. So, LCM(12, 18) = (2 × 2) × (3 × 3) = 4 × 9 = 36
---
5. Now, find the product of the numbers: 12 × 18 = 216
---
6. Finally, find the product of HCF and LCM: HCF(12, 18) × LCM(12, 18) = 6 × 36 = 216
---
ANSWER: Since 12 × 18 = 216 and HCF(12, 18) × LCM(12, 18) = 216, the relationship is verified.

Why It Matters

This concept is super useful in fields like Computer Science for optimizing algorithms and in Cryptography for securing online transactions, like when you pay with UPI. Engineers use it to design efficient systems, and data scientists apply similar logic for managing large datasets. Knowing this helps build a strong base for future tech careers!

Common Mistakes

MISTAKE: Applying the rule for more than two numbers. | CORRECTION: This specific rule (HCF × LCM = Product of Numbers) only works for two positive numbers. It does not hold true for three or more numbers.

MISTAKE: Confusing HCF and LCM values during multiplication. | CORRECTION: Always ensure you are multiplying the HCF by the LCM, not by one of the original numbers or by itself. Double-check your calculated HCF and LCM before applying the formula.

MISTAKE: Forgetting to find the product of the original numbers. | CORRECTION: The rule compares the product of HCF and LCM with the product of the original numbers. Make sure you calculate both sides of the equation to verify the relationship.

Practice Questions

Try It Yourself

QUESTION: The HCF of two numbers is 8 and their LCM is 48. If one number is 16, what is the other number? | ANSWER: 24

QUESTION: The product of two numbers is 1800. If their HCF is 15, what is their LCM? | ANSWER: 120

QUESTION: Can the HCF of two numbers be 10 and their LCM be 45? Explain why or why not. | ANSWER: No. The HCF must always be a factor of the LCM. Here, 10 is not a factor of 45 (45 divided by 10 is not a whole number). So, this combination is not possible.

MCQ

Quick Quiz

If the HCF of two numbers is 5 and their product is 150, what is their LCM?

750

The Correct Answer Is:

According to the relationship, HCF × LCM = Product of Numbers. So, 5 × LCM = 150. Dividing 150 by 5 gives LCM = 30.

Real World Connection

In the Real World

Imagine you're a computer programmer building a system to schedule buses. You might need to find the HCF and LCM of different bus routes' timings to ensure they don't clash or to find the next time all buses meet at a central depot. This concept helps in optimizing routes and making sure everything runs smoothly, just like how traffic lights are timed in busy Indian cities.

Key Vocabulary

Key Terms

HCF (Highest Common Factor): The largest number that divides two or more numbers exactly. | LCM (Lowest Common Multiple): The smallest positive number that is a multiple of two or more numbers. | Product: The result obtained when two or more numbers are multiplied together. | Prime Factors: Prime numbers that multiply together to make the original number.

What's Next

What to Learn Next

Great job understanding this! Next, you can explore how to find HCF and LCM for three or more numbers, or dive into more complex problems involving fractions and decimals. This foundational knowledge will make those topics much easier to grasp, helping you solve even bigger math challenges!