What is Prime Factorisation Method for LCM?

S3-SA4-0035

Grade Level:

Class 6

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

Definition

What is it?

The Prime Factorisation Method for LCM is a way to find the Least Common Multiple (LCM) of two or more numbers by breaking them down into their prime factors. This method helps us find the smallest number that is a multiple of all the given numbers.

Simple Example

Quick Example

Imagine you have two friends, Rohan and Priya. Rohan visits the library every 3 days, and Priya visits every 4 days. If they both visited today, when will they both visit the library together again? We need to find the LCM of 3 and 4 to know this.

Worked Example

Step-by-Step

Let's find the LCM of 12 and 18 using the Prime Factorisation Method.

Step 1: Find the prime factors of 12.
12 = 2 x 6
12 = 2 x 2 x 3

---

Step 2: Find the prime factors of 18.
18 = 2 x 9
18 = 2 x 3 x 3

---

Step 3: List all the prime factors from both numbers. For each prime factor, take the highest power (the maximum number of times it appears in any single factorisation).
Prime factors of 12: 2^2 x 3^1
Prime factors of 18: 2^1 x 3^2

---

Step 4: Identify the highest power for each unique prime factor.
For prime factor '2', the highest power is 2^2 (from 12).
For prime factor '3', the highest power is 3^2 (from 18).

---

Step 5: Multiply these highest powers together to get the LCM.
LCM = 2^2 x 3^2
LCM = (2 x 2) x (3 x 3)
LCM = 4 x 9
LCM = 36

Answer: The LCM of 12 and 18 is 36.

Why It Matters

Understanding LCM is super important in many fields! Engineers use it when designing gears or circuits to make sure parts move together smoothly. Computer scientists use it in algorithms, and even in data science, it helps manage data efficiently. It's a foundational concept for problem-solving.

Common Mistakes

MISTAKE: Multiplying all prime factors together without considering their highest powers. For example, for LCM of 12 (2x2x3) and 18 (2x3x3), multiplying 2x2x3x2x3x3. | CORRECTION: For each unique prime factor, choose its highest power (the maximum times it appears in any one number's prime factorisation) and then multiply those highest powers.

MISTAKE: Confusing LCM with HCF. Students sometimes find common factors instead of taking all factors with their highest powers. | CORRECTION: Remember, HCF looks for COMMON factors with the LOWEST powers. LCM looks for ALL unique factors with their HIGHEST powers.

MISTAKE: Not breaking numbers down into only prime factors. For example, writing 12 as 2 x 6 and stopping there. | CORRECTION: Always break down numbers completely until all factors are prime numbers (like 2, 3, 5, 7, etc.).

Practice Questions

Try It Yourself

QUESTION: Find the LCM of 10 and 15 using the Prime Factorisation Method. | ANSWER: 30

QUESTION: What is the LCM of 8, 12, and 20? Show your steps. | ANSWER: 120

QUESTION: Two traffic lights at a busy Mumbai junction change every 45 seconds and 60 seconds, respectively. If they both change at 9:00 AM, when will they next change simultaneously? Use the Prime Factorisation Method. | ANSWER: 9:03 AM (LCM is 180 seconds, which is 3 minutes)

MCQ

Quick Quiz

What is the LCM of 6 and 9 using the Prime Factorisation Method?

The Correct Answer Is:

Prime factors of 6 are 2 x 3. Prime factors of 9 are 3 x 3 (or 3^2). The highest power of 2 is 2^1, and the highest power of 3 is 3^2. So, LCM = 2 x 3^2 = 2 x 9 = 18.

Real World Connection

In the Real World

This method helps in planning! For example, if two different bus routes from a Delhi bus stand depart every 15 minutes and 20 minutes respectively, you can use LCM to find out when both buses will depart at the same time again. This helps in scheduling and managing public transport efficiently.

Key Vocabulary

Key Terms

PRIME FACTOR: A prime number that divides a given number exactly. | LEAST COMMON MULTIPLE (LCM): The smallest positive number that is a multiple of two or more given numbers. | MULTIPLE: The result of multiplying a number by an integer. | PRIME NUMBER: A whole number greater than 1 that has exactly two divisors: 1 and itself.

What's Next

What to Learn Next

Great job learning about LCM! Now that you understand prime factorisation, you can explore the 'Highest Common Factor (HCF)' using the same method. HCF is another important concept that uses prime factors to find the largest number that divides two or more numbers exactly.