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What is the LCM by Division 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 Division Method is a simple way to find the smallest number that is a multiple of two or more given numbers. We do this by dividing the numbers together by their common prime factors until no more common factors exist.
Simple Example
Quick Example
Imagine you have two friends, Rahul and Priya. Rahul visits your house every 4 days, and Priya visits every 6 days. If they both visited today, when will they both visit together again? The LCM of 4 and 6 will tell us the answer. Using the division method, we find it's 12 days.
Worked Example
Step-by-Step
Let's find the LCM of 12 and 18 using the Division Method.
Step 1: Write down the numbers side-by-side: 12, 18
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Step 2: Find the smallest prime number that divides at least one of them. Here, 2 divides both 12 and 18. Divide them:
2 | 12, 18
| 6, 9
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Step 3: Look at the new numbers (6, 9). Again, find the smallest prime number that divides at least one. Here, 3 divides both 6 and 9. Divide them:
2 | 12, 18
3 | 6, 9
| 2, 3
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Step 4: Look at the new numbers (2, 3). No common prime factor divides both. Now, divide by the remaining prime factors individually until you get 1.
2 | 12, 18
3 | 6, 9
2 | 2, 3
| 1, 3
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Step 5: Now divide by 3 for the remaining number:
2 | 12, 18
3 | 6, 9
2 | 2, 3
3 | 1, 3
| 1, 1
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Step 6: Multiply all the prime divisors from the left column: 2 x 3 x 2 x 3 = 36.
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Answer: The LCM of 12 and 18 is 36.
Why It Matters
Understanding LCM is super important! In computer science, it helps in scheduling tasks efficiently, like how your mobile apps update. In engineering, it's used to design gears or ensure different parts move together smoothly. Even data scientists use it for pattern recognition in large datasets, like predicting cricket match outcomes!
Common Mistakes
MISTAKE: Not using only prime numbers for division. | CORRECTION: Always divide by prime numbers (2, 3, 5, 7, etc.) only. Don't use composite numbers like 4 or 6.
MISTAKE: Stopping too early when only one number is divisible. | CORRECTION: Continue dividing even if only one number is divisible by a prime factor. Bring down the non-divisible number as it is.
MISTAKE: Forgetting to multiply all the prime factors found. | CORRECTION: The LCM is the product of ALL the prime factors you used for division, including those that divided only one number.
Practice Questions
Try It Yourself
QUESTION: Find the LCM of 8 and 10 using the Division Method. | ANSWER: 40
QUESTION: What is the LCM of 9, 15, and 20? | ANSWER: 180
QUESTION: Two traffic lights at a crossing change every 30 seconds and 45 seconds respectively. If they both change at 9:00 AM, when will they next change simultaneously? (Hint: Find LCM of 30 and 45). | ANSWER: 90 seconds (1 minute 30 seconds) later, at 9:01:30 AM.
MCQ
Quick Quiz
Which of these is the correct first step when finding the LCM of 15 and 25 using the division method?
Divide both numbers by 3.
Divide both numbers by 5.
Divide both numbers by 15.
Divide 15 by 3 and 25 by 5.
The Correct Answer Is:
B
The correct first step is to divide by the smallest common prime factor. Both 15 and 25 are divisible by 5, which is a prime number.
Real World Connection
In the Real World
Think about planning your day with multiple tasks! If you need to charge your phone every 8 hours and water your plants every 12 hours, finding the LCM helps you figure out when you'll do both at the same time. This is similar to how scientists at ISRO plan satellite orbits or how app developers schedule notifications on your phone to avoid clashes.
Key Vocabulary
Key Terms
LCM: Least Common Multiple, the smallest number that is a multiple of two or more numbers. | Prime Factor: A prime number that divides a given number exactly. | Multiple: The result of multiplying a number by an integer. | Division Method: A technique for finding LCM by dividing numbers by common prime factors.
What's Next
What to Learn Next
Great job understanding LCM! Next, you should explore HCF (Highest Common Factor) or GCD (Greatest Common Divisor). HCF is closely related to LCM and learning it will give you a complete picture of how numbers share factors and multiples.
