What is LCM by Prime Factorisation Method? | Simple Explanation for Class 6

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What is the LCM by Prime Factorisation Method?

Grade Level:

Class 6

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

Definition

What is it?

The LCM (Least Common Multiple) by Prime Factorisation Method is a way to find the smallest number that is a multiple of two or more given numbers. We do this by breaking down each number into its prime factors and then multiplying the highest powers of all unique prime factors together.

Simple Example

Quick Example

Imagine you have two friends, Rohan and Priya, who visit your chai stall. Rohan visits every 3 days, and Priya visits every 4 days. If they both visited today, when will they both visit your chai stall again at the same time? We need to find the LCM of 3 and 4.

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
So, 12 = 2^2 x 3^1

---Step 2: Find the prime factors of 18.
18 = 2 x 9
18 = 2 x 3 x 3
So, 18 = 2^1 x 3^2

---Step 3: List all unique prime factors from both numbers. The unique prime factors are 2 and 3.

---Step 4: For each unique prime factor, take the highest power it appears in either factorisation.
For prime factor 2: The powers are 2^2 (from 12) and 2^1 (from 18). The highest power is 2^2.
For prime factor 3: The powers are 3^1 (from 12) and 3^2 (from 18). The highest power is 3^2.

---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

So, the LCM of 12 and 18 is 36.

Why It Matters

Understanding LCM helps engineers schedule tasks efficiently, like when different parts of a machine need maintenance. In computer science, it's used in algorithms for data processing and even in cryptography to keep information secure. It's a foundational skill for many tech careers!

Common Mistakes

MISTAKE: Multiplying all prime factors without considering powers (e.g., for LCM of 6 and 8, doing 2 x 3 x 2 x 2 x 2 instead of 2^3 x 3) | CORRECTION: Always take the highest power of each unique prime factor present in any of the numbers.

MISTAKE: Confusing LCM with HCF (Highest Common Factor) and only taking common prime factors. | CORRECTION: For LCM, you must include ALL unique prime factors from all numbers, taking their highest powers.

MISTAKE: Missing a prime factor when breaking down a number (e.g., thinking 12 = 2 x 6, and stopping there). | CORRECTION: Continue factorising until all factors are prime numbers. A prime number can only be divided by 1 and itself (like 2, 3, 5, 7).

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 9, 12, and 15? | ANSWER: 180

QUESTION: Two traffic lights at a busy Mumbai junction change after every 40 seconds and 60 seconds, respectively. If they both change at 7:00 AM, when will they next change simultaneously? Find the LCM of 40 and 60 to solve this. | ANSWER: 120 seconds (or 2 minutes), so they will next change at 7:02 AM.

MCQ

Quick Quiz

Which of the following is the correct prime factorisation for finding the LCM of 8 and 12?

8 = 2 x 4, 12 = 2 x 6

8 = 2^3, 12 = 2^2 x 3^1

8 = 2 x 2 x 2, 12 = 3 x 4

8 = 2^2 x 2, 12 = 2 x 3 x 2

The Correct Answer Is:

Option B correctly shows the prime factorisation for both numbers, breaking them down completely into their prime factors with powers. The other options either have non-prime factors or incomplete prime factorisations.

Real World Connection

In the Real World

Imagine you're a data scientist working for a delivery app like Zepto or Swiggy. You might use LCM to schedule optimal delivery routes, figuring out when different delivery partners will complete their loops to ensure efficient pick-ups and drops, minimising wait times for customers.

Key Vocabulary

Key Terms

PRIME NUMBER: A whole number greater than 1 that has only two factors: 1 and itself (e.g., 2, 3, 5) | FACTORISATION: Breaking down a number into a product of its factors | MULTIPLE: The result of multiplying a number by an integer (e.g., multiples of 3 are 3, 6, 9) | LEAST COMMON MULTIPLE (LCM): The smallest positive number that is a multiple of two or more numbers | POWER: How many times a number is multiplied by itself (e.g., 2^3 means 2 x 2 x 2)

What's Next

What to Learn Next

Great job learning LCM! Next, explore the HCF (Highest Common Factor) by Prime Factorisation Method. Understanding both LCM and HCF will give you a strong foundation for solving many real-world math problems!