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What is the Prime Factorisation Method for LCM?
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. It 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 6 days, and Priya visits every 8 days. If they both visit today, to find out when they will visit together again, you'd use LCM. The LCM of 6 and 8 is 24, so they will meet again in 24 days.
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
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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
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Step 3: List all the prime factors that appear in either number. These are 2 and 3.
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Step 4: For each prime factor, take the highest power it appears with 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.
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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
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Answer: The LCM of 12 and 18 is 36.
Why It Matters
Understanding LCM is crucial in many fields, from scheduling tasks in computer science to designing circuits in engineering. Data scientists use it to find patterns, and even in cryptography, understanding number properties is key to secure communication.
Common Mistakes
MISTAKE: Students often confuse LCM with HCF and pick the lowest power of common factors. | CORRECTION: For LCM, you need to take ALL prime factors that appear in ANY number, and for each, take its HIGHEST power.
MISTAKE: Missing a prime factor that appears in only one of the numbers. | CORRECTION: Remember to include all prime factors, even if they only appear in the factorisation of one number, and still use their highest power.
MISTAKE: Incorrectly listing prime factors (e.g., writing 4 as a prime factor). | CORRECTION: Always break down numbers completely until only prime numbers (like 2, 3, 5, 7, etc.) are left.
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 different types of LED lights blink at intervals of 8 seconds and 10 seconds respectively. If they blink together at 7:00 AM, when will they blink together again? | ANSWER: 7:00 AM and 40 seconds
MCQ
Quick Quiz
Which of these is the correct first step when finding the LCM of 20 and 30 using prime factorisation?
Divide both numbers by 10
List the multiples of 20 and 30
Find the prime factors of 20 and 30 separately
Multiply 20 by 30
The Correct Answer Is:
C
The Prime Factorisation Method starts by breaking down each number into its prime factors. Options A, B, and D are not the correct first step for this specific method.
Real World Connection
In the Real World
Imagine a traffic signal in a busy Indian city. If two signals turn green at different intervals, say 45 seconds and 60 seconds, engineers use LCM to figure out when both signals will turn green at the exact same moment again. This helps in traffic management and planning.
Key Vocabulary
Key Terms
Prime Factor: A prime number that divides a given number exactly. | Factorisation: The process of breaking down a number into its prime factors. | Multiple: A number that can be divided by another number without a remainder. | Least Common Multiple (LCM): The smallest positive integer that is a multiple of two or more numbers.
What's Next
What to Learn Next
Great job understanding LCM! Next, you can explore the 'Relationship between HCF and LCM'. This will show you a cool formula that connects both concepts and makes solving some problems even faster. Keep up the good work!
